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Find a function given the volume of the solid of revolution it generates

by RoRi

Let f(x) be a function continuous on an interval [0,a]. The volume of the solid of revolution obtained by rotating f about the x-axis on the interval [0,a] is given by

    \[ V = a^2 + a \]

for every a > 0. Find a formula for the function f.

Using the formula for the volume of the solid of revolution generated by a function on an interval we know

    \[ V = \pi \int_0^a (f(t))^2 \, dt \quad \implies \quad a^2 + a = \pi \int_0^a (f(t))^2 \, dt. \]

Now we differentiate both sides of this equation using the fundamental theorem of calculus on the right-hand side,

    \begin{align*} x^2 + x = \pi \int_0^x (f(t))^2 \, dt && \implies && 2x + 1 = \pi (f(x))^2 \\[9pt] && \implies && f(x) = \sqrt{\frac{2x+1}{\pi}}. \end{align*}

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