Multiple integral(tripple integral)
Download as PPTX, PDF18 likes6,402 views
This document discusses triple integrals and their evaluation in different coordinate systems. It begins by defining a triple integral as the generalization of a double integral to three dimensions. It then discusses evaluating triple integrals in rectangular, cylindrical, and spherical coordinate systems. For each system it provides the coordinate transformations between rectangular and the other system and the formula for the volume element used in the triple integral.
More Related Content
Similar to Multiple integral(tripple integral) (20)
Multiple integral(tripple integral)
- 1. L. D. College Of Engineering, Ahmedabad
- 2. Calculus Multiple Integrals - Triple Integrals
- 3. Index:- Triple Integrals Triple Integrals in Cylindrical Co- ordinates Triple Integrals in Spherical Co- ordinates Change of order of Integration Jacobian of several variables
- 4. Triple Integrals: The triple integral is defined in a similar manner to that of the double integral if f(x,y,z) is continuous and single-valued function of x, y, z over the region R of space enclosed by the surface S. We sub divide the region R into rectangular cells by planes parallel to the three co-ordinate planes(fig 1).The parallelopiped cells may have the dimensions of δx, δy and δz.We number the cells inside R as δV1, δV2,…..δVn.
- 5. . In each such parallelopiped cell we choose an arbitrary point in the k th pareallelopiped cell whose volume is δVk and then we form the sum =
- 6. .
- 7. .
- 8. .
- 9. .
- 10. Triple Integrals In Cylindrical Coordinates: We obtain cylindrical coordinates for space by combining polar coordinates (r, θ) in the xy-plane with the usual z-axis. This assigns every point in space one or more coordinates triples of the form (r, θ, z) as shown in figure.2.
- 11. .
- 12. Definition : Cylindrical coordinate Cylindrical coordinate represent a point P in space by orders triples (r, θ, z) in which 1. (r, θ) are polar coordinates for the vertical projection of P on xy-plane. 2. z is the rectangular vertical coordinates. The rectangular (x , y , z) and cylindrical coordinates are related by the usual equations as follow : x = r cosθ, y = r sinθ , z = z 𝑟2 = 𝑥2 + 𝑦2 , tanθ = 𝑦 𝑥 .
- 13. Formula for tripple integral in cylindrical coordinates where,volume element in cylindrical coordinates is given by dV = rdzdrdθ
- 14. Triple Integrals in Spherical Co- ordinates: Spherical coordinates locate points in space is with two angles and one distance, as shown in figure.3. The first coordinate P = |OP|, is the point’s distance from the origin. The second coordinate ф, is the angle OP make with the positive z-axis. It is required to lie in the interval 0 ≤ ф ≤ π. The third coordinate is the angle θ as measured in cylindrical coordinates.
- 15. Figure.3
- 16. Definition : Spherical coordinates Spherical coordinates represent a point P in ordered triples (ƍ , θ , ф) in which 1. ƍ is the distance from P to the origin. 2. θ is the angle from cylindrical coordinates. 3. ф is the angle OP makes with the positive z-axis (0 ≤ ф ≤ π). The rectangular coordinates (x , y, z) and spherical coordinates are related by the following equations : x = ƍ sinф cosθ , y = ƍ sinф sinθ, z = P cosф.
- 17. Formula for Triple integral in spherical coordinates:- where, D = {(ƍ , θ , ф) | a ≤ ƍ ≤ b, α ≤ θ ≤ β, c ≤ ф ≤ d} and dV = ƍ2 sin фdƍdф.