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Compute the volume of a solid with given properties

by RoRi

Given a solid with square cross sections perpendicular to the x-axis and with their center on the x-axis. If the cross section square at x has side length 2x^2, compute the volume of the solid for 0 \leq x \leq a.

The area of the cross sections is given by

    \[ A(x) = (2x^2)^2 = 4x^4. \]

So, the volume of the solid for 0 \leq x \leq a is given by

    \[ V = \int_0^a 4x^4 \, dx = \frac{4}{5}a^5. \]

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