# Thread: find the volume of the solid by rotating..

1. ## find the volume of the solid by rotating..

hi guys, having problems with this practice question.

Find the volume of the solid obtained by rotating the region under the graph of $\displaystyle y = \sqrt{x}$ between x = 2 and x = 7 around the x-axis.

thank u for help

EDIT: i rotated the graph of $\displaystyle y = \sqrt{x}$ then i grabed the intervals 2 till 7. so is the integration $\displaystyle \int_2^{7} (x^{1/2})^2$

2. Originally Posted by jvignacio
hi guys, having problems with this practice question.

Find the volume of the solid obtained by rotating the region under the graph of $\displaystyle y = \sqrt{x}$ between x = 2 and x = 7 around the x-axis.

thank u for help

EDIT: i rotated the graph of $\displaystyle y = \sqrt{x}$ then i grabed the intervals 2 till 7. so is the integration $\displaystyle \int_2^{7} (x^{1/2})^2$
The volume of revolution can be envisaged as approximatly composed of disks of radius $\displaystyle \sqrt{x}$ and thickness $\displaystyle \delta x$ , and hence volume of each disk is $\displaystyle \pi |x| \delta x$. Then the volume of the solid of revolution is approximatly the sum of the volumes of the disks comprising the solid, or in the limit the integral:

$\displaystyle V=\int_2^7 \pi |x|\ dx$

and as $\displaystyle x$ is positive over the range of integration we can drop the $\displaystyle |.|$ to get:

$\displaystyle V=\int_2^7 \pi x\ dx$

CB

3. Originally Posted by CaptainBlack
The volume of revolution can be envisaged as approximatly composed of disks of radius $\displaystyle \sqrt{x}$ and thickness $\displaystyle \delta x$ , and hence volume of each disk is $\displaystyle \pi |x| \delta x$. Then the volume of the solid of revolution is approximatly the sum of the volumes of the disks comprising the solid, or in the limit the integral:

$\displaystyle V=\int_2^7 \pi |x|\ dx$

and as $\displaystyle x$ is positive over the range of integration we can drop the $\displaystyle |.|$ to get:

$\displaystyle V=\int_2^7 \pi x\ dx$

CB
so now i just integrate $\displaystyle V=\int_2^7 \pi x\ dx$ ?

4. Originally Posted by jvignacio
so now i just integrate $\displaystyle V=\int_2^7 \pi x\ dx$ ?
Yes.

CB

5. Originally Posted by CaptainBlack
Yes.

CB
$\displaystyle V=\int_2^7 \pi x\ dx$

= 70.6858347 correct?

6. Originally Posted by jvignacio
$\displaystyle V=\int_2^7 \pi x\ dx$

= 70.6858347 correct?
You should leave it as $\displaystyle \frac{45 \pi}{2}$

CB