 Aleah Booth

2022-07-23

How to calculate the volume of cos(x) around the x-axis and the y-axis separately?
I have difficulties finding the right formula how to calculate the volume rotating cos(x) around the x-axis and y-axis separately can you please give me a hint how to do it?
The interval is $x=\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ Caylee Davenport

Expert

Step 1
1) Around the x-axis, you get
${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\pi \left(\mathrm{cos}x{\right)}^{2}dx=2{\int }_{0}^{\frac{\pi }{2}}\pi \left(\mathrm{cos}x{\right)}^{2}dx$,
using the Disc method (and symmetry).
Step 2
2) Around the y-axis, you get ${\int }_{0}^{\frac{\pi }{2}}2\pi x\mathrm{cos}x\phantom{\rule{thickmathspace}{0ex}}dx$, using the Shell method.
(Notice that the right half of the region generates the whole solid in this case.) Leila Jennings

Expert

Step 1
First draw the function. We know that Volume is given by ${\int }_{a}^{b}\pi f\left(x{\right)}^{2}dx$. Therefore ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\pi {\mathrm{cos}}^{2}\left(x\right)dx$ and that's an easy integral.
Step 2
While integrating along y axis your function will change to ${\mathrm{cos}}^{-1}x$ and hence integral will become ${\int }_{0}^{1}\pi \left({\mathrm{cos}}^{-1}\left(y\right){\right)}^{2}dy$

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