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 . 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 2rh.)

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.

5.compare each of these methods.