Calculating Volume of Revolution: Solving for Unknowns Using Equations

[delsoo]

https://www.physicsforums.com/attachments/69284

Homework Statement

i have done the part a, for b , i use the key in the (circled part equation ) into calculator .. my ans is also different form the ans given. is my concept correct by the way?

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

One obvious point is that you are missing a factor of "\pi". The area of a circle is \pi r^2= \pi y^2.

 

[delsoo]

after adding pi, my ans is 2.80... the ans is 5.047746784, which part is wrong?

 

[STEMucator]

It would be a good idea here to use vertical line segments, otherwise you're going to have to set up multiple integrals. So leave everything as $y(x)$, then:

$r_{in} = 0$
$r_{out} = 1 + \frac{1}{4x^2 + 1}$
$height = dx$

$dV = 2\pi(\frac{r_{in} + r_{out}}{2})(r_{out} - r_{in})(height)$

Integrating the volume element should give you the answer you want.

 

[Saitama]

after adding pi, my ans is 2.80... the ans is 5.047746784, which part is wrong?

delsoo, your method is fine and I seem to get the same definite integral as you (which gives the correct answer too). The definite integral you have to evaluate is:

$$\pi\int_0^{1/2} \left(1+\frac{1}{4x^2+1}\right)^2\,dx$$

If you drop the factor of $\pi$, you should get 1.60675.

Use the substitution $2x=\tan\theta$ to make things easier.

 

[STEMucator]

delsoo, your method is fine and I seem to get the same definite integral as you (which gives the correct answer too). The definite integral you have to evaluate is:

$$\pi\int_0^{1/2} \left(1+\frac{1}{4x^2+1}\right)^2\,dx$$

If you drop the factor of $\pi$, you should get 1.60675.

Erm this is misleading.

The answer is indeed 5.04775 complements of wolfram:

http://www.wolframalpha.com/input/?i=integrate+2pi%28+%281%2B+1%2F%284x^2%2B1%29%29%2F2+%29%281%2B+1%2F%284x^2%2B1%29%29+from+0+to+1%2F2

 

[Saitama]

Erm this is misleading.

Can you please explain to me how my statements are misleading? :)

 

[STEMucator]

Can you please explain to me how my statements are misleading? :)

Your integrand is fine, it's just the answer you got I was worried about.

 

[Saitama]

Your integrand is fine, it's just the answer you got I was worried about.

I must be missing something but what is the problem with the answer I wrote? Are you talking about "1.60675"?

 

[STEMucator]

I must be missing something but what is the problem with the answer I wrote? Are you talking about "1.60675"?

Yeah I wasn't sure why you wrote that.

 

[Saitama]

Yeah I wasn't sure why you wrote that.

Ah, I think I worded it poorly. What I meant was this:
$$\int_0^1 \left(1+\frac{1}{4x^2+1}\right)^2\,dx=1.60675$$
And I feel delsoo did some mistake while evaluating the above definite integral.