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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).

06_applications_of_the_integral-64.gif

Figure 6.2.14

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

06_applications_of_the_integral-65.gif

The hemisphere has volume

06_applications_of_the_integral-66.gif

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

06_applications_of_the_integral-67.gif

The hemisphere has volume

06_applications_of_the_integral-68.gif

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

½V=06_applications_of_the_integral-69.gif=06_applications_of_the_integral-70.gif=06_applications_of_the_integral-71.gif=⅔π3

Thus again V = (4/3)πr3.


Last Update: 2006-11-22