SOLUTION TO PROBLEM R-10

The volume generated is given by the integral

$$\int_0^b\pi\left (f(x)\right )^2\; dx$$

So, for all b>0, we have the identity

$$b^2=\int_0^b\pi\left (f(x)\right )^2\; dx$$

Take the derivative of both sides, using the Fundamental Theorem of Calculus. Then

$$2b=\pi\left (f(b)\right )^2$$

So $f(b)=\sqrt{2b/\pi}$. In other words,

$$f(x)=\frac{\sqrt{2\pi x}}{\pi}$$

Questions and/or comments should be directed to Judy Downey or Griff Elder

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Last modified:   Fri Mar 22 13:12:44 CST 2002