# Volumes of revolution

#### Petrus

##### Well-known member
Calculate the volumes of the rotation bodies which arises when the area D in the xy-plane bounded by x-axis and curve $$\displaystyle 7x-x^2$$may rotate around x- respective y-axes.
I will calculate $$\displaystyle V_x$$ and $$\displaystyle V_y$$ I start to get crit point $$\displaystyle x_1=0$$ and $$\displaystyle x_2=7$$
rotate on y-axe:
$$\displaystyle 2\pi\int_a^bf(x)dx$$
so we get $$\displaystyle 2\pi[\frac{7x^2}{2}-\frac{x^3}{3}]_0^7$$ $$\displaystyle V_y=\frac{2\pi*343}{6}$$
rotate on x axe:
$$\displaystyle \pi\int_a^bf(x)^2dx$$
so we start with:$$\displaystyle (7x-x^2)^2=49x^2-14x^3+x^4$$ so we get $$\displaystyle [\frac{49x^3}{3}-\frac{14x^4}{4}+\frac{x^5}{5}]_0^7$$ that means $$\displaystyle V_x=\frac{16087\pi}{30}$$ What I am doing wrong?

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
First , why do you think you are doing something wrong ?

#### MarkFL

Staff member
I was going to ask the same thing...are you expecting you should get the same volume with a different axis of rotation? This in only true with the axes you are given for a particular family of parabolas, and this one is not in that family. See this topic:

http://www.mathhelpboards.com/f35/problem-week-37-december-10th-2012-a-2714/

I believe that problem was inspired by a problem I helped you with in the past.

Your formula for the shell method (revolving about the $y$-axis) is missing the radius of the shell. Your other formula for the disk method (revolving about the $x$-axis) is correct.

#### Petrus

##### Well-known member
I was going to ask the same thing...are you expecting you should get the same volume with a different axis of rotation? This in only true with the axes you are given for a particular family of parabolas, and this one is not in that family. See this topic:

http://www.mathhelpboards.com/f35/problem-week-37-december-10th-2012-a-2714/

I believe that problem was inspired by a problem I helped you with in the past.

Your formula for the shell method (revolving about the $y$-axis) is missing the radius of the shell. Your other formula for the disk method (revolving about the $x$-axis) is correct.
Well its a programe we put our answer on so we see if we get correct or wrong what do you mean missing the radius of the shell?

#### MarkFL

Staff member
...what do you mean missing the radius of the shell?
The volume of an arbitrary shell is:

$$\displaystyle dV=2\pi rh\,dx$$

where:

$$\displaystyle h=f(x)$$

You also need to write $r$ in terms of $x$. Do you see how your formula is missing the radius?

#### Petrus

##### Well-known member
The volume of an arbitrary shell is:

$$\displaystyle dV=2\pi rh\,dx$$

where:

$$\displaystyle h=f(x)$$

You also need to write $r$ in terms of $x$. Do you see how your formula is missing the radius?
Yes I do, I did think wrong when I try use my brain(and some memory) for the formula. $$\displaystyle 2\pi\int_0^7x(7x-x^2)$$ is this correct now?

Edit: got correct answer! Thanks MarkFL and ZaidAylafey!

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