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! Last edited: