# Thread: Volumes of revolution

1. ## Volumes of revolution

Hello all,
New here but needing some help on a few questions in involving areas no in the first quadrant.
Question 1: Find the volume of the solid generated by revolving the region bounded by y= -x^2-1 and y=x^2-2x-5 about the line y=11.

Question 2: is using the same regions about x=4. The bounded region is in quadrants 3&4 and this seems to be throwing me off.
Thanks in advance for your help.
ninelegs

2. ## Re: Volumes of revolution

Originally Posted by ninelegs
Hello all,
New here but needing some help on a few questions in involving areas no in the first quadrant.
Question 1: Find the volume of the solid generated by revolving the region bounded by y= -x^2-1 and y=x^2-2x-5 about the line y=11.

Question 2: is using the same regions about x=4. The bounded region is in quadrants 3&4 and this seems to be throwing me off.
Thanks in advance for your help.
ninelegs
start by making a sketch of the curves relative to the rotation axis

find where the curves intersect ...

$-x^2-1 = x^2-2x-5$

$x = 2 , x = -1$

method of washers ...

$R(x) = 11-(x^2-2x-5) = 16+2x-x^2$

$r(x) = 11 - (-x^2-1) = x^2+12$

$V = \pi \int_{-1}^2 [R(x)]^2 - [r(x)]^2 \, dx$

about $x = 4$, use the method of cylindrical shells ...

$V = 2\pi \int_a^b \cdot r(x) \cdot h(x) \, dx$

$r(x) = 4-x$

$h(x) = (-x^2-1)-(x^2-2x-5)$

3. ## Re: Volumes of revolution

Thank you so much! I kept adding the distance from the axis of revolution and it was screwing me up.
I have just one more for you.
Find the volume of the solid generated by revolving the region bounded by the y-axis, the line y=1 and the curve x= y^2-4y about the line x=3.

thanks again

4. ## Re: Volumes of revolution

Originally Posted by ninelegs
Thank you so much! I kept adding the distance from the axis of revolution and it was screwing me up.
I have just one more for you.
Find the volume of the solid generated by revolving the region bounded by the y-axis, the line y=1 and the curve x= y^2-4y about the line x=3.

thanks again
you make a sketch and set this one up ... it will be w/r to y