# Thread: V of solid rev..........

1. ## V of solid rev..........

Hi,
I have done the question on the picture attached
I don't believe it could be the correct answer, can someone help me please

Thanks

2. actually should my answer be 1114?

3. Originally Posted by wolfhound Hi,
I have done the question on the picture attached
I don't believe it could be the correct answer, can someone help me please

Thanks
$\displaystyle (x^2+4x)^2=x^4+8x^3+16x^2$

4. Silly me,
still wrong I think

5. Originally Posted by wolfhound Can someone help me with this please.....
That's correct to the nearest whole number.

6. but is it a rule to make it into a fraction and keep the pi(sign) or is it ok to do like I did

7. $\displaystyle V = \pi \int_0^3 (x^2+4x)^2 \, dx$

$\displaystyle V = \pi \int_0^3 x^4 + 8x^3 + 16x^2 \, dx$

$\displaystyle V = \pi \left[\frac{x^5}{5} + 2x^4 + \frac{16x^3}{3}\right]_0^3$

$\displaystyle V = \pi \left[\frac{3^5}{5} + 2 \cdot 3^4 + \frac{16 \cdot 3^3}{3}\right]$

$\displaystyle V = \frac{1773\pi}{5} \approx 1114$

8. Originally Posted by wolfhound Is anyone there to help me with this annoying solid of revolution??????????
Hi wolfhound.

It always helps to understand how the situation looks on a diagram.

The volume of revolution is the sum of the areas of a series of "wafer-thin" discs (an "infinite" sum of them, evaluated as their widths "deta-x" goes to zero).

The radius of the discs in this case is f(x) because the curve is being rotated around the x-axis.

the area of a disc is $\displaystyle {\pi}r^2={\pi}[f(x)]^2$

All of these disc areas are integrated from x=0 to x=3.
This gives the volume of revolution.

You need to get used to how to do this notationally,
as your written work needs to be redone.

$\displaystyle Disc\ area={\pi}\left(x^2+4x\right)^2={\pi}\left(x^4+8x^ 3+16x^2\right)$

Therefore, the volume of revolution is

$\displaystyle \int_{x=0}^{x=3}{\pi}\left(x^4+8x^3+16x^2\right)dx$

$\displaystyle ={\pi}\left[\frac{x^5}{5}+\frac{8x^4}{4}+\frac{16x^3}{3}\right]$ from x=0 to 3

I've shown the disc from an angle (hence it looks like an ellipse!)

9. ## Thanks for the help, I must make more sense when I write it out..

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