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Divanshu's question via email about a volume by revolution

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
volume question.JPG

Here is a graph of the region to be rotated. Notice that it is being rotated around the same line that is the lower boundary.

volume graph.JPG

The volume will be exactly the same if everything is moved down by 4 units, with the advantage of being rotated around the x-axis. So using the rule for finding the volume of a solid formed by rotating $\displaystyle \begin{align*} f(x) \end{align*}$ around the x axis: $\displaystyle \begin{align*} V = \int_a^b{ \pi\,\left[ f(x) \right] ^2\,\mathrm{d}x } \end{align*}$ the volume we want is $\displaystyle \begin{align*} V &= \int_0^8{\pi\, \left( x^2 + 4 \right) ^2 \,\mathrm{d}x } \end{align*}$.