Introduction to Engineering Mathematics



Vector Calculus for Engineers

CME100, Fall 2004

Handout #3

Double and Triple Integrals in Cartesian, Cylindrical, and Spherical Coordinates with Applications

1. Calculate the volume under the plane [pic] over the rectangular region R: [pic]

2. Find the volume of the prism whose base is the triangle in the xy-plane bounded by the x-axis and the lines [pic] and [pic], and whose top lies in the plane [pic]

3. Calculate [pic] over the triangle in the xy-plane bounded by the x-axis, [pic], and [pic]

4. A thin plate covers the triangular region bounded by the x-axis and the lines [pic] and [pic] in the first quadrant. The density is given by [pic]. Find the mass and the center of mass of the plate.

5. For the region in the previous example find the moment of inertia [pic]

6. Find the center of mass of a semi-circular disk bounded by the x-axis and [pic]

a) using Cartesian coordinates

b) using polar coordinates

7. Using triple integrals, find the volume of the region enclosed by the plane [pic] and coordinate planes [pic], [pic], and [pic]

8. Find the center of mass of a solid of constant density [pic] bounded by the disk [pic] in the plane [pic] and above by the paraboloid [pic]

9. Find moments of inertia [pic], [pic] and [pic] for the rectangular solid of constant density [pic] and dimensions in the x, y and z directions of a, b and c respectively.

10. Using cylindrical coordinates find the volume and the center of mass of a cone with uniform density formed by [pic] and [pic]

11. Using cylindrical coordinates find the volume above the xy-plane bounded by [pic] and [pic]

12. Using spherical coordinates find the center of mass of a hemisphere of radius R

13. Find the moment of inertia of a uniform solid sphere of mass M and radius R

14. Using transformation of coordinates find the area enclosed by [pic], [pic], [pic] and [pic]

15. Find the Jacobian of transformation between:

a) Cartesian and polar coordinates

b) Cartesian and cylindrical coordinates

c) Cartesian and spherical coordinates

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