Calculus | Differential and Integral Calculus Last Updated : 07 Apr, 2025 Comments Improve Suggest changes Like Article Like Report Calculus was founded by Newton and Leibniz. Calculus is a branch of mathematics that helps us study change. It is used to understand how things change over time or how quantities grow, shrink, or accumulate. There are two main parts of calculus:Differential Calculus: It helps us calculate the rate of change of one quantity concerning another. This rate of change is called the derivative.Example: Finding how fast a balloon inflates as you pump air into it.Calculating the slope of a hill (steepness).Integral Calculus: helps us calculate the total accumulation of change. This accumulation is called the integral.Example: Calculating the area under a curve (e.g., finding the distance traveled by a car when you know its speed at every moment).Determining the total rainfall collected in a reservoir.Note: The process of finding the value of a derivative is called differentiation, and the process of finding the value of an integral is called integration.Basic of CalculusThis section covers the basics of calculus, including functions, limits, and continuity. You will learn key techniques for finding limits and understanding discontinuities in functions.FunctionsDomain and Range of a FunctionLimitsFormal Definition of LimitsOne-Sided LimitsInfinite LimitsLimits at InfinityTechniques for Finding LimitsLimits by RationalizationLimits using Algebraic ManipulationEstimating Limits from GraphsEstimating Limits from TablesSqueeze TheoremLimits of Trigonometric FunctionsProperties of LimitsContinuity of FunctionsContinuity at a PointIntermediate Value TheoremExtreme Value TheoremDiscontinuityTypes of DiscontinuityDifferential CalculusThis section explores differential calculus, focusing on derivatives and their applications. You will learn differentiation rules, including the power, product, quotient, and chain rules, along with real-life applications such as rate of change, extrema, and curve sketching.DifferentiabilityMean Value Theorem: Cauchy's Mean Value TheoremRolle's Mean Value TheoremDerivativeDerivative by First PrincipleAlgebra of Derivative of FunctionsRules for DifferentiationPower RuleProduct RuleQuotient RuleChain ruleFormulas for DifferentiationImplicit DifferentiationLogarithmic DifferentiationParametric DifferentiationExamples of DerivativesDerivatives of Polynomial FunctionsDerivatives of Trigonometric FunctionsDerivatives of Exponential FunctionsDerivatives of Logarithmic FunctionsDerivatives of Inverse Trigonometric FunctionsDerivatives of Inverse FunctionsDerivatives of Composite FunctionsApplication of DerivativesCritical PointsRate Change of QuantitiesIncreasing FunctionDecreasing FunctionApproximationMaxima or Minima: Absolute Minima and Maxima | Relative Minima and MaximaTangent and NormalConcavity and Points of InflectionCurve SketchingPartial DerivativesHigher Order DerivativesAntiderivativesReal-Life Application of DifferentiationIntegral CalculusThis section covers the fundamentals of integral calculus, exploring the concept of integration and its relationship to differentiation. You will learn various methods of integration, such as substitution and integration by parts, and apply these techniques to solve real-world problems involving areas, volumes, and surfaces.Introduction to IntegrationAntiderivative: Integration as an Inverse Process of DifferentiationFundamental Theorem of CalculusTypes of IntegralsDefinite IntegralsDefinite Integral as the Limit of a Riemann SumProperties of Definite IntegralsEvaluation of Definite IntegralsIndefinite IntegralsImproper IntegralsRiemann SumRiemann Sums in Summation NotationFunctions defined by IntegralsIntegration FormulasMethods of IntegrationIntegration by SubstitutionIntegration by Trigonometric SubstitutionIntegration by PartsIntegration by Partial FractionApplication of IntegrationArea Under a CurveArea Between CurvesArea Between Polar CurvesVolume of Solids of RevolutionArc Length of CurvesSurface Area of RevolutionLine IntegralSurface IntegralDouble IntegrationTriple IntegralDifferential EquationsThis section introduces differential equations, covering their types, including ordinary and partial differential equations, and methods for solving them. You will explore key concepts such as order, degree, and techniques like exact and separable equations, with a focus on first and second-order differential equations.Introduction to Differential EquationsOrder and Degree of Differential Equations to Differential EquationsTypes of Differential EquationsExact Differential EquationsSeparable Differential EquationsOrdinary Differential EquationsPartial Differential EquationsLinear Differential EquationsHomogeneous Differential EquationsFirst Order Differential EquationSecond Order Differential EquationAlso, Check Calculus Cheat Sheet.Practice for CalculusThis section provides a series of practice quizzes and questions to reinforce your understanding of key calculus concepts. You'll test your knowledge on limits, continuity, maxima and minima, and integration through interactive exercises.Limits - QuizContinuity of Function - QuizMaxima and Minima - QuizIntegration - QuizPractice Questions on CalculusPrograms for CalculusThis section offers practical programming solutions for implementing calculus operations. You’ll learn how to write efficient code in Python and MATLAB, enhancing your skills in applying mathematical concepts through programming.Calculus with PythonProgram to Differentiate Given PolynomialProgram to Calculate Double IntegrationCalculus in MATLABDiffrentiation in MATLAB Comment More infoAdvertise with us Next Article Limits in Calculus abhishek1 Follow Improve Article Tags : Mathematics School Learning Tutorials Calculus Maths-Categories +1 More Similar Reads Calculus | Differential and Integral Calculus Calculus was founded by Newton and Leibniz. Calculus is a branch of mathematics that helps us study change. It is used to understand how things change over time or how quantities grow, shrink, or accumulate. There are two main parts of calculus:Differential Calculus: It helps us calculate the rate o 4 min read LimitsLimits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp 12 min read Formal Definition of LimitsLimit of a function f(x) at a particular point x, describes the behaviour of that function very close to that particular point. It does not give the value of the function at that particular point, it just gives the value function that seems to be taking on at that point.The figure below presents a s 6 min read Properties of LimitsIn mathematics, a limit is a concept that describes how a function or sequence behaves as its input gets closer to a certain value. For example, if you keep dividing 1 by larger and larger numbers (like 1/2, 1/3, 1/4, etc.), the result gets closer and closer to 0. The limit of this sequence as the d 6 min read Indeterminate FormsAssume a function F(x)=\frac{f(x)}{g(x)} which is undefined at x=a but it may approach a limit as x approaches a. The process of determining such a limit is known as evaluation of indeterminate forms. The L' Hospital Rule helps in the evaluation of indeterminate forms. According to this rule- \lim_{ 3 min read Strategy in Finding LimitsLimits have been really useful in the field of calculus, they become a solid foundation for defining many concepts like continuity, differentiability, integrals, and derivatives. These concepts further help us analyze a lot of functions and their behavior in calculus. Limits have been the foundation 9 min read Determining Limits using Algebraic ManipulationLimits give us the power to approximate functions and see the values they are approaching. Limit is not the value of the function at a particular point. It is the value which the function is approaching as one moves towards the given point. There are many ways to solve the limits, often limits are e 7 min read Limits of Trigonometric FunctionsTrigonometry is one of the most important branches of Mathematics. We know that there are six trigonometric functions, and the limit of a trigonometric function is the limit taken to each trigonometric function. We can easily find the limit of trigonometric functions, and the limit of the trigonomet 9 min read Limits by Direct SubstitutionLimits are building blocks of calculus. They are the values that a function seems to be taking on while we reach a particular point. They help in calculating the rate of change of the functions. The concept of derivatives has been defined with limits. They also help us define the concepts of continu 7 min read Estimating Limits from GraphsThe concept of limits has been around for thousands of years. Earlier mathematicians in ancient civilizations used limits to approximate the area of a circle. However the formal concept was not around till the 19th century. This concept is essential to calculus and serves as a building block for ana 9 min read Estimating Limits from TablesLimits tell us a lot about function behavior. They help mathematicians and engineers reason about the function their behavior and their properties. They form the basis for almost every important concept in calculus. Limits help us estimate the values function seems to be taking at a particular point 9 min read Sandwich TheoremSandwich Theorem, also known as the Squeeze Theorem, is a fundamental concept in calculus used to find the limit of a function. The theorem works by "sandwiching" a given function between two other functions whose limits are easier to determine. If the two bounding functions converge to the same lim 6 min read Continuity and DifferentiabilityAlgebra of Continuous Functions - Continuity and Differentiability | Class 12 MathsAlgebra of Continuous Functions deals with the utilization of continuous functions in equations involving the varied binary operations you've got studied so. We'll also mention a composition rule that may not be familiar to you but is extremely important for future applications.Since the continuity 6 min read Continuity and Discontinuity in CalculusContinuity and Discontinuity: Continuity and discontinuity are fundamental concepts in calculus and mathematical analysis, describing the behavior of functions. A function is continuous at a point if you can draw the graph of the function at that point without lifting your pen from the paper. Contin 7 min read DerivativesDerivatives | First and Second Order Derivatives, Formulas and ExamplesA derivative is a concept in mathematics that measures how a function changes as its input changes. For example:If you're driving a car, the derivative of your position with respect to time is your speed. It tells you how fast your position is changing as time passes.If you're looking at a graph of 8 min read Differentiation FormulasDifferentiation Formulas: Differentiation allows us to analyze how a function changes over its domain. We define the process of finding the derivatives as differentiation. The derivative of any function ?(x) is represented as d/dx.?(x)In this article, we will learn about various differentiation form 7 min read Algebra of Derivative of FunctionsDerivatives are an integral part of calculus. They measure the rate of change in any quantity. Suppose there is a water tank from which water is leaking. A local engineer is asked to measure the time in which the water tank will become empty. In such a scenario, the engineer needs to know two things 8 min read Product Rule FormulaProduct rule is a fundamental principle in calculus used to differentiate the product of two functions. The rule states that the derivative of a product of two differentiable functions is given as, "product of first function with the derivative of second function added with differentiation of first 8 min read Quotient Rule: Formula, Proof, Definition, ExamplesQuotient Rule is a method for finding the derivative of a function that is the quotient of two other functions. It is a method used for differentiating problems where one function is divided by another. We use the quotient rule when we have to find the derivative of a function of the form: f(x)/g(x) 6 min read Power RulePower Rule is a fundamental rule in the calculation of derivatives that helps us find the derivatives of functions with exponents. Exponents can take any form, including any function itself. With the help of the Power Rule, we can differentiate polynomial functions, functions with variable exponents 6 min read Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f 11 min read Derivatives of Various FunctionsDerivatives of Polynomial FunctionsDerivatives are used in Calculus to measure the rate of change of a function with respect to a variable. The use of derivatives is very important in Mathematics. It is used to solve many problems in mathematics like to find out maxima or minima of a function, the slope of a function, or determining 3 min read Derivatives of Trigonometric FunctionsThe derivative of a function f(x) is the rate at which the value of the function changes when the input is changed. In this context, x is called the independent variable, and f(x) is called the dependent variable. Derivatives have applications in almost every aspect of our lives. From rocket launche 8 min read Derivatives of Inverse FunctionsIn mathematics, a function(e.g. f), is said to be an inverse of another(e.g. g), if given the output of g returns the input value given to f. Additionally, this must hold true for every element in the domain co-domain(range) of g. E.g. assuming x and y are constants if g(x) = y and f(y) = x then the 11 min read Derivatives of Inverse Trigonometric FunctionsDerivatives of Inverse Trigonometric Functions: Every mathematical function, from the simplest to the most complex, has an inverse. In mathematics, the inverse usually means the opposite. In addition, the inverse is subtraction. For multiplication, it's division. In the same way for trigonometric fu 14 min read Derivatives of Implicit Functions - Continuity and Differentiability | Class 12 MathsImplicit functions are functions where a specific variable cannot be expressed as a function of the other variable. A function that depends on more than one variable. Implicit Differentiation helps us compute the derivative of y with respect to x without solving the given equation for y, this can be 6 min read Implicit DifferentiationImplicit Differentiation is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function. For example, we need to find the slope of a circle with an origin at 0 and a radius r. Its equation is given as x2 + y2 = r2. Now, to find the s 5 min read Implicit differentiation - Advanced ExamplesIn the previous article, we have discussed the introduction part and some basic examples of Implicit differentiation. So in this article, we will discuss some advanced examples of implicit differentiation. Table of Content Implicit DifferentiationMethod to solveImplicit differentiation Formula Solve 5 min read Derivatives of Composite FunctionsDerivatives are an essential part of calculus. They help us in calculating the rate of change, maxima, and minima for the functions. Derivatives by definition are given by using limits, which is called the first form of the derivative. We already know how to calculate the derivatives for standard fu 6 min read Chain Rule: Theorem, Formula and Solved ExamplesThe Chain Rule is a way to find the derivative of composite functions. It is one of the basic rules used in mathematics for solving differential equations. It helps us to find the derivative of composite functions such as (3x2 + 1)4, (sin 4x), e3x, (ln x)2, and others. Only the derivatives of compos 8 min read Derivative of Exponential FunctionsDerivative of Exponential Function stands for differentiating functions expressed in the form of exponents. We know that exponential functions exist in two forms, ax where a is a real number r and is greater than 0 and the other form is ex where e is Euler's Number and the value of e is 2.718 . . . 7 min read Advanced DifferentiationLogarithmic DifferentiationMethod of finding a function's derivative by first taking the logarithm and then differentiating is called logarithmic differentiation. This method is specially used when the function is type y = f(x)g(x). In this type of problem where y is a composite function, we first need to take a logarithm, ma 8 min read Proofs for the derivatives of eË£ and ln(x) - Advanced differentiationIn this article, we are going to cover the proofs of the derivative of the functions ln(x) and ex. Before proceeding there are two things that we need to revise: The first principle of derivative Finding the derivative of a function by computing this limit is known as differentiation from first prin 3 min read Derivative of Functions in Parametric FormsParametric Differentiation refers to the differentiation of a function in which the dependent and independent variables are equated to a third variable. Derivatives of the functions express the rate of change in the functions. We know how to calculate the derivatives for standard functions. Chain ru 7 min read Disguised Derivatives - Advanced differentiation | Class 12 MathsThe dictionary meaning of “disguise†is “unrecognizableâ€. Disguised derivative means “unrecognized derivativeâ€. In this type of problem, the definition of derivative is hidden in the form of a limit. At a glance, the problem seems to be solvable using limit properties but it is much easier to solve 6 min read Second Order Derivatives: Rules , Formula and Examples (Class 12 Maths)The Second Order Derivative is defined as the derivative of the first derivative of the given function. The first-order derivative at a given point gives us the information about the slope of the tangent at that point or the instantaneous rate of change of a function at that point. Second-Order Deri 10 min read Higher Order DerivativesHigher order derivatives refer to the derivatives of a function that are obtained by repeatedly differentiating the original function.The first derivative of a function, f′(x), represents the rate of change or slope of the function at a point.The second derivative, f′′(x), is the derivative of the f 6 min read Application of DerivativesApplication of DerivativesDerivative of a variable y with respect to x is defined as the ratio between the change in y and the change in x, depending upon the condition that changes in x should be very small tending towards zero. dy/dx = lim ∆x⇢0 ∆y / ∆x = lim h⇢0 (f(x + h) - f(x)) / h.Where, ∆x OR h is change in x, and∆y OR 14 min read Mean Value TheoremThe Mean Value Theorem states that for a curve passing through two given points there exists at least one point on the curve where the tangent is parallel to the secant passing through the two given points. Mean Value Theorem is abbreviated as MVT. This theorem was first proposed by an Indian Mathem 12 min read Rolle's Mean Value TheoremRolle's theorem one of the core theorem of calculus states that, for a differentiable function that attains equal values at two distinct points then it must have at least one fixed point somewhere between them where the first derivative of the function is zero.Rolle's Theorem and the Mean Value Theo 8 min read Lagrange's Mean Value TheoremLagrange's Mean Value Theorem (LMVT) is a fundamental result in differential calculus, providing a formalized way to understand the behavior of differentiable functions. This theorem generalizes Rolle's Theorem and has significant applications in various fields of engineering, physics, and applied m 9 min read Rolle's Theorem and Lagrange's Mean Value TheoremRolle's Theorem and Lagrange's Mean Value Theorem: Mean Value Theorems (MVT) are the basic theorems used in mathematics. They are used to solve various types of problems in Mathematics. Mean Value Theorem is also called Lagrenges's Mean Value Theorem. Rolle’s Theorem is a subcase of the mean value t 11 min read Advanced DifferentiationDerivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of 8 min read Tangents and NormalsTangent and Normals are the lines that are used to define various properties of the curves. We define tangent as the line which touches the circle only at one point and normal is the line that is perpendicular to the tangent at the point of tangency. Any tangent of the curve passing through the poin 13 min read Equation of Tangents and NormalsDerivatives are used to find rate of change of a function with respect to variables. To find rate of change of function with respect to a variable differentiating it with respect to that variable is required. Rate of change of function y = f(x) with respect to x is defined by dy/dx or f'(x). For exa 6 min read Increasing and Decreasing FunctionsIncreasing and decreasing functions refer to the behavior of a function's graph as you move from left to right along the x-axis. A function is considered increasing if for any two values x1 and x2​ such that x1 < x2 ​, the function value at x1​ is less than the function value at x2​ (i.e., f( x1) 13 min read Increasing and Decreasing IntervalsIncreasing and decreasing intervals are the intervals of real numbers in which real-valued functions are increasing and decreasing respectively. Derivatives are a way of measuring the rate of change of a variable.Increasing and Decreasing IntervalsWhen it comes to functions and calculus, derivatives 10 min read Concave FunctionGraphs of the functions give us a lot of information about the nature of the function, the trends, and the critical points like maxima and minima of the function. Derivatives allow us to mathematically analyze these functions and their sign can give us information about the maximum and minimum of th 9 min read Inflection PointInflection Point describes a point where the curvature of a curve changes direction. It represents the transition from a concave to a convex shape or vice versa. Let's learn about Inflection Points in detail, including Concavity of Function and solved examples. Table of Content Inflection Point Defi 9 min read Curve SketchingCurve Sketching as its name suggests helps us sketch the approximate graph of any given function which can further help us visualize the shape and behavior of a function graphically. Curve sketching isn't any sure-shot algorithm that after application spits out the graph of any desired function but 14 min read Absolute Minima and MaximaAbsolute Maxima and Minima are the maximum and minimum values of the function defined on a fixed interval. A function in general can have high values or low values as we move along the function. The maximum value of the function in any interval is called the maxima and the minimum value of the funct 11 min read Relative Minima and MaximaRelative maxima and minima are the points defined in any function such that at these points the value of the function is either maximum or minimum in their neighborhood. Relative maxima and minima depend on their neighborhood point and are calculated accordingly. We find the relative maxima and mini 8 min read IntegralsIntegralsIntegrals: An integral in mathematics is a continuous analog of a sum that is used to determine areas, volumes, and their generalizations. Performing integration is the process of computing an integral and is one of the two basic concepts of calculus.Integral in Calculus is the branch of Mathematics 11 min read Antiderivative: Integration as Inverse Process of DifferentiationAn antiderivative is a function that reverses the process of differentiation. It is also known as the indefinite integral. If F(x) is the antiderivative of f(x), it means that:d/dx[F(x)] = f(x)In other words, F(x) is a function whose derivative is f(x).Antiderivatives include a family of functions t 6 min read Differentiation and Integration FormulaDifferentiation and Integration are two mathematical operations used to find change in a function or a quantity with respect to another quantity instantaneously and over a period, respectively. Differentiation is an instantaneous rate of change, and it breaks down the function for that instant with 11 min read Functions Defined by IntegralsWhile thinking about functions, we always imagine that a function is a mathematical machine that gives us an output for any input we give. It is usually thought of in terms of mathematical expressions like squares, exponential and trigonometric function, etc. It is also possible to define the functi 5 min read Indefinite IntegrationIndefinite IntegralsIntegrals are also known as anti-derivatives as integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-different 6 min read Integration FormulasIntegration Formulas are the basic formulas used to solve various integral problems. They are used to find the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, and logarithmic and exponential functions. These integration formulas are beneficial for finding 10 min read Integration by Substitution MethodIntegration by substitution or u-substitution is a highly used method of finding the integration of a complex function by reducing it to a simpler function and then finding its integration. Suppose we have to find the integration of f(x) where the direct integration of f(x) is not possible. So we su 7 min read Integration by Partial FractionsIntegration by Partial Fractions is one of the methods of integration, which is used to find the integral of the rational functions. In Partial Fraction decomposition, an improper-looking rational function is decomposed into the sum of various proper rational functions.If f(x) and g(x) are polynomia 8 min read Partial Fraction ExpansionIf f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime 8 min read Integration by PartsIntegration is the calculation of an integral which is used in maths for finding many useful quantities such as areas, volumes, displacement, etc., that occur due to a collection of small data which cannot be measured singularly. Integrals are denoted using " ∫ ".Integration NotationThe concept of i 6 min read Integration by U-substitutionFinding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind 7 min read Reverse Chain RuleIntegrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve 6 min read Trigonometric Substitution: Method, Formula and Solved ExamplesTrigonometric substitution is a process in which the substitution of a trigonometric function into another expression takes place. It is used to evaluate integrals or it is a method for finding antiderivatives of functions that contain square roots of quadratic expressions or rational powers of the 6 min read Integration of Trigonometric FunctionsIntegration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below.Below is an example of the Integra 9 min read Definite IntegrationDefinite Integral | MathematicsDefinite integrals are the extension after indefinite integrals, definite integrals have limits [a, b]. It gives the area of a curve bounded between given limits. \int_{a}^{b}F(x)dx , It denotes the area of curve F(x) bounded between a and b, where a is the lower limit and b is the upper limit. Note 1 min read Computing Definite IntegralsIntegrals are a very important part of the calculus. They allow us to calculate the anti-derivatives, that is given a function's derivative, integrals give the function as output. Other important applications of integrals include calculating the area under the curve, the volume enclosed by a surface 5 min read Fundamental Theorem of Calculus | Part 1, Part 2Fundamental Theorem of Calculus is the basic theorem that is widely used for defining a relation between integrating a function of differentiating a function. The fundamental theorem of calculus is widely useful for solving various differential and integral problems and making the solution easy for 11 min read Finding Derivative with Fundamental Theorem of CalculusIntegrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit 5 min read Evaluating Definite IntegralsIntegration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function 8 min read Properties of Definite IntegralsProperties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as \int_{a}^{b}f(x) = F(b) − F(a)There are many properties regarding definite integral. We will discuss each property one by one with proof.Defin 7 min read Definite Integrals of Piecewise FunctionsImagine a graph with a function drawn on it, it can be a straight line or a curve, or anything as long as it is a function. Now, this is just one function on the graph. Can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight lin 9 min read Improper IntegralsImproper integrals are definite integrals where one or both of the boundaries are at infinity or where the Integrand has a vertical asymptote in the interval of integration. Computing the area up to infinity seems like an intractable problem, but through some clever manipulation, such problems can b 5 min read Riemann SumsRiemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int 7 min read Riemann Sums in Summation NotationRiemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These 8 min read Definite Integral as the Limit of a Riemann SumDefinite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li 7 min read Trapezoidal RuleThe Trapezoidal Rule is a fundamental method in numerical integration used to approximate the value of a definite integral of the form b∫a f(x) dx. It estimates the area under the curve y = f(x) by dividing the interval [a, b] into smaller subintervals and approximating the region under the curve as 12 min read Like