Multiple Integral

Integration is a mathematical tool used to calculate areas, volumes, mass, and many other quantities. When a function depends on more than one variable, multiple integrals are used to calculate the total value over a region.

• A single integral adds up small pieces of a function along one variable.
• A double integral adds up small pieces over a two-dimensional area.
• A triple integral adds up small pieces over a three-dimensional space.

Multiple integrals are useful in various fields, including physics, engineering, and statistics. They help to compute volume, mass, center of mass, probability, and other important quantities.

Types of Multiple Integrals

There are mainly three types of integrals based on the number of variables involved:

Single Integral - A single integral is used for a function of one variable.

Example - \int_{a}^{b} f(x) \, dx

Double Integral - A double integral is used for a function of two variables, such as f(x,y). It is used to calculate volume under a surface or total mass over a region.

Example - \iint_{R} f(x, y) \, dA = \int_{y_1}^{y_2} \int_{x_1}^{x_2} f(x, y) \, dx \, dy

Triple Integral - A triple integral is used for functions of three variables, such as f(x, y, z). It is used to calculate total mass, charge, or volume in three-dimensional space.

Example - \iiint_{V} f(x, y, z) \, dV = \int_{z_1}^{z_2} \int_{y_1}^{y_2} \int_{x_1}^{x_2} f(x, y, z) \, dx \, dy \, dz

Applications of Multiple Integrals

• Calculating the volume of irregular objects.
• Finding the mass of objects with variable density.
• Computing probabilities in statistics.
• Engineering applications such as electric charge distribution and moment of inertia.

Properties of Multiple Integrals,

1) Linearity - \iint_R [a f(x,y) + b g(x,y)] \, dA = a \iint_R f(x,y) \, dA + b \iint_R g(x,y) \, dA

2) Additive Inverse - If the region R is divided into two subregions, R₁ and R₂

\iint_R f(x,y) \, dA = \iint_{R_1} f(x,y) \, dA + \iint_{R_2} f(x,y) \, dA

3) Monotonicity - If f(x,y) and g(x ,y) for all points in R

\iint_R f(x,y) \, dA \le \iint_R g(x,y) \, dA

These properties also extend to triple integrals, but are most easily explained using double integrals.

Steps to Solve a Multiple Integral

• Identify the function and the region of integration
• Set up the integral with correct limits.
• Integrate the innermost variable first.
• Continue integrating outward, one variable at a time.
• The final value is the total area, volume, or quantity.

Solved Example on Multiple Integral

Example 1: Find the area of the triangular region bounded by x = 0, y = 0, and x + y = 1 using a double integral.

Solution:

Set up an integral

\iint_R 1 \, dA = \int_0^1 \int_0^{1-x} 1 \, dy \, dx

Integrate with respect to y first:

\int_0^{1-x} 1 \, dy = 1 - x

Next, integrate with respect to x :

\int_0^1 (1 - x) \, dx = \frac{1}{2}

The area of the triangular region is 1/2

Example 2: Find the Mass of a Rectangular Plate with Density 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, density p(x,y) = x + y

Solution:

Set up the double integral for mass:

Mass = \iint_R \rho(x, y) \, dA = \int_0^3 \int_0^2 (x + y) \, dx \, dy

Integrate with respect to x first:

\int_0^2 (x + y) \, dx = \left[ \frac{x^2}{2} + xy \right]_0^2 = 2 + 2y

Next, integrate with respect to y :

\int_0^3 (2 + 2y) \, dy = \left[ 2y + y^2 \right]_0^3 = 6 + 9 = 15

Example 3: Evaluate the triple integral:

\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} (x + y + z) \, dz\, dy\, dx

Solution:

Step 1 - Integrate w.r.t z

\int_{0}^{1} (x + y + z)\, dz = \left[ x z + y z + \frac{z^2}{2} \right]_0^1 = [x(1) + y(1) + \frac{1^2}{2}] - [x(0) + y(0) + \frac{0^2}{2}] = x + y + \frac{1}{2}

Step 2 - Integrate w.r.t y

\int_{0}^{1} (x + y + \frac{1}{2})\, dy = \left[ x y + \frac{y^2}{2} + \frac{1}{2} y \right]_0^1= [x(1) + \frac{1^2}{2} + \frac{1}{2}(1)] - [x(0) + \frac{0^2}{2} + \frac{1}{2}(0)] = x + 1

Step 3 - Integrate w.r.t x

\int_{0}^{1} (x + 1)\, dx = \left[ \frac{x^2}{2} + x \right]_0^1 = \frac{1^2}{2} + 1 - (0 + 0) = \frac{3}{2}

I = 3/2

Unsolved Question on Multiple Integral

Question 1: Evaluate the double integral of f(x,y) = 2x + 3y over the rectangle defined by 0 ≤ x ≤ 20, 0 ≤ y ≤ 30.

Question 2: Evaluate the double integral of f(x,y) = xy over the triangular region bounded by x = 0 and x + y = 2.

Question 3:Evaluate the triple integral of f(x,y,z) = x + y + z over the region bounded by z = 0 and z = 2 − x − y, 0x ≥ 0.

Question 4:Evaluate the triple integral of f(x,y,z) = xyz over the cube 0 ≤ x, y, z ≤ 1.