ScholarMatic: Explanation & Answer
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Volume of a Sphere Five Ways
Consider the sphere centered at the origin with radius R. From volume formulas in high school, you know that this sphere has volume 4 3 ?R3 . You will show this is true using integrals. For each part, make a sketch showing either the disks/shells or the variables of integration.
1. The sphere can be thought of as a solid of revolution about either the x, y, or z axis.
(a) Using the disk method, find the volume of the sphere as a solid of revolution. (Recall that the area of a disk of radius r is ?r2 .)
(b) Using the shell method, find the volume of the sphere as a solid of revolution. (Recall that the lateral surface area of a cylinder with radius r and height h is 2?rh.)
2. Use a triple integral in Cartesian coordinates to find the volume of the sphere.
3. Use a triple integral in Cylindrical coordinates to find the volume of the sphere.
4. Use a triple integral in Spherical coordinates to find the volume of the sphere.
ScholarMatic: Explanation & Answer
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