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

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

Answer & Explanation

Caylee Davenport

Expert

2022-07-24Added 14 answers

Step 1

1) Around the x-axis, you get

${\int}_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\pi (\mathrm{cos}x{)}^{2}dx=2{\int}_{0}^{\frac{\pi}{2}}\pi (\mathrm{cos}x{)}^{2}dx$,

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.)

1) Around the x-axis, you get

${\int}_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\pi (\mathrm{cos}x{)}^{2}dx=2{\int}_{0}^{\frac{\pi}{2}}\pi (\mathrm{cos}x{)}^{2}dx$,

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 ${\int}_{a}^{b}\pi f(x{)}^{2}dx$. Therefore ${\int}_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\pi {\mathrm{cos}}^{2}(x)dx$ 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 ${\int}_{0}^{1}\pi ({\mathrm{cos}}^{-1}(y){)}^{2}dy$

First draw the function. We know that Volume is given by ${\int}_{a}^{b}\pi f(x{)}^{2}dx$. Therefore ${\int}_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\pi {\mathrm{cos}}^{2}(x)dx$ 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 ${\int}_{0}^{1}\pi ({\mathrm{cos}}^{-1}(y){)}^{2}dy$

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