# Homework Help: Volume of revolution

1. May 1, 2014

### delsoo

View attachment 69284 1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

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Last edited: May 1, 2014
2. May 1, 2014

### HallsofIvy

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

3. May 1, 2014

### delsoo

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

4. May 1, 2014

### Zondrina

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.

5. May 1, 2014

### Saitama

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.

6. May 1, 2014

### Zondrina

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

7. May 1, 2014

### Saitama

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

8. May 1, 2014

9. May 1, 2014

### Saitama

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

10. May 1, 2014

### Zondrina

Yeah I wasn't sure why you wrote that.

11. May 1, 2014

### Saitama

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.