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Two volume-related questions

Yuuki

Member
Jun 7, 2013
43
#1 describe the solid obtained by

\pi \int_{0}^{\pi/2}\cos ^2xdx

i thought this meant that the area function is

pi \cos ^2xdx and since \cos ^2xdx = 1^2 - \sinx ^2xdx
it's the solid obtained by rotating region between y = sinx and y = 0 for 0 <= x <= pi / 2.

but this was wrong.
how?

#2 compute the volume of a torus which has inner radius R and outer radius R + r.

no idea.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: two volume-related questions

#1 describe the solid obtained by

\pi \int_{0}^{\pi/2}\cos ^2xdx

i thought this meant that the area function is

pi \cos ^2xdx and since \cos ^2xdx = 1^2 - \sinx ^2xdx
it's the solid obtained by rotating region between y = sinx and y = 0 for 0 <= x <= pi / 2.

but this was wrong.
how?
The formula for the disk method of computing the volume of a solid of revolution about the $x$-axis is:

\(\displaystyle V=\pi\int_a^b f^2(x)\,dx\)

Do you see now how to describe the region being rotated?

#2 compute the volume of a torus which has inner radius R and outer radius R + r.

no idea.
I would direct you to this thread which should give you an indication of how to proceed:

http://mathhelpboards.com/questions...ahoo-answers-regarding-volume-torus-7992.html
 

Yuuki

Member
Jun 7, 2013
43
Re: two volume-related questions

i think i now get #1


but i don't know how your link relates to my question.
is the solid in question in the thread a torus??
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: two volume-related questions

...
but i don't know how your link relates to my question.
is the solid in question in the thread a torus??
Yes, the names of the variables are different, but it is a torus. I just wanted you to see the method used.

To work the second problem using the variables given, I would recommend revolving a circle of radius $\dfrac{r}{2}$ and centered at $\left(R+\dfrac{r}{2},0 \right)$ about the $y$-axis. Do you see how this will create a torus whose inner radius is $R$ and whose outer radius is $R+r$?