Example 5

Derive the formula V = (4/3)πr³ for the volume of a sphere by both the Disc Method and the Cylindrical Shell Method.

The circle of radius r and center at the origin has the equation

x² + y² = r².

The region R inside this circle in the first quadrant will generate a hemisphere of radius r when it is rotated about the x-axis (Figure 6.2.14).

Figure 6.2.14

First take x as the independent variable and use the Disc Method. R is the region under the curve

The hemisphere has volume

Therefore the sphere has volume

V = (4/3)πr³

Now take y as the independent variable and use the Cylindrical Shell Method. R is the region under the curve

The hemisphere has volume

Putting u = r² - y², du = - 2y dy, we get

½V====⅔π³

Thus again V = (4/3)πr³.