Aleah Booth

Answered

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 $x=\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$

Answer & Explanation

Caylee Davenport

Expert

2022-07-24Added 14 answers

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

2022-07-25Added 5 answers

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 ${\int }_{0}^{1}\pi \left({\mathrm{cos}}^{-1}\left(y\right){\right)}^{2}dy$

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?