Triple integrals and applications
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1. Triple integrals are used to find the volume of a solid region in 3D space. They are defined similarly to double integrals, encompassing the region with a network of boxes and taking the limit of Riemann sums. 2. Triple integrals can be evaluated using one of six orders of integration. The limits of integration are determined by first finding the innermost limits, which may depend on the outer variables, and then projecting the region onto different coordinate planes. 3. As an example, the center of mass of a unit cube is found by using a density function proportional to the square of the distance from the origin. Any order of integration can be used due to the symmetry of the region.
Triple integrals and applications
- 1. 1 Triple Integrals and Applications Roll Number- 16BELR046
- 2. 2 Triple Integrals The procedure used to define a triple integral follows that used for double integrals. Consider a function f of three variables that is continuous over a bounded solid region Q. Then, encompass Q with a network of boxes and form the inner partition consisting of all boxes lying entirely within Q, as shown in Figure.
- 3. 3 Triple Integrals The volume of the ith box is The norm of the partition is the length of the longest diagonal of the n boxes in the partition. Choose a point (xi, yi, zi) in each box and form the Riemann sum
- 4. 4 Triple Integrals Taking the limit as leads to the following definition.
- 5. 5 Triple Integrals ∫ ∫ ∫ f (x, y, z) dV can be evaluated with an iterated integral using one of the six possible orders of integration:
- 7. 7 Example1–EvaluatingaTripleIteratedIntegral Evaluate the triple iterated integral Solution: For the first integration, hold x and y constant and integrate with respect to z.
- 8. 8 Example 1 – Solution For the second integration, hold x constant and integrate with respect to y. Finally, integrate with respect to x. cont’d
- 9. 9 Triple Integrals To find the limits for a particular order of integration, it is generally advisable first to determine the innermost limits, which may be functions of the outer two variables. Then, by projecting the solid Q onto the coordinate plane of the outer two variables, you can determine their limits of integration by the methods used for double integrals.
- 10. 10 Triple Integrals For instance, to evaluate first determine the limits for z, and then the integral has the form
- 11. 11 Triple Integrals By projecting the solid Q onto the xy-plane, you can determine the limits for x and y as you did for double integrals, as shown in Figure.
- 12. 12 Example5–FindingtheCenterofMassofaSolidRegion Find the center of mass of the unit cube shown in Figure , given that the density at the point (x, y, z) is proportional to the square of its distance from the origin.
- 13. 13 Example 5 – Solution Because the density at (x, y, z) is proportional to the square of the distance between (0, 0, 0) and (x, y, z), you have You can use this density function to find the mass of the cube. Because of the symmetry of the region, any order of integration will produce an integral of comparable difficulty.
- 14. 14 Example 5 – Solution cont’d