# Thread: using integrals to find volume

1. ## using integrals to find volume

i have this problem where i have to find the volume of the solid obtained by rotationg the region bounded by the given curves about the specified line.

the specified line is x=1
im also given x=y^2 and x=1. the question is, when i integrate do i make it:
(1-1)^2 -(1-y^2) dy? im unsure how to set this one up

2. Originally Posted by gbux512
i have this problem where i have to find the volume of the solid obtained by rotationg the region bounded by the given curves about the specified line.

the specified line is x=1
im also given x=y^2 and x=1. the question is, when i integrate do i make it:
(1-1)^2 -(1-y^2) dy? im unsure how to set this one up
If you are bounding the graph from x=0 to x=1, then rotating that part of the graph about x=1,
then integrate in the y-direction, note that you are integrating the discs formed from rotating
that part of the curve about x=1.

The radii of these circles is $1-x=1-y^2$

Hence $\displaystyle\int{{\pi}r^2}dy=\int_{y=-1}^{y=1}{\pi}\left(1-y^2\right)^2dy$

3. ok so moving down the line of problems, i have one more thats giving me trouble, y=x, y=0 x=2 x=4, about x=1

so i start off taking the integral from 2-4 of (1-y)^2 -(1-0)^2 dy

then i foil it out to get 1-2y+y^2 -1 dy

then i integrate it *after ditching the 1's) and get -y^2 + (y^3)/3 |2,4

sooo im now at -4^2 + (4^3)/3- (2^2+ (2^3)/3)

which simplifies out to 48/3 + 64/3 + -12/3 -8/3 =92(pi)/3

this is obviously not the answer in the back of the book, but after picking my brain for HOURS i cant find my flaw :/

4. Originally Posted by gbux512
ok so moving down the line of problems, i have one more thats giving me trouble, y=x, y=0 x=2 x=4, about x=1

so i start off taking the integral from 2-4 of (1-y)^2 -(1-0)^2 dy

then i foil it out to get 1-2y+y^2 -1 dy

then i integrate it *after ditching the 1's) and get -y^2 + (y^3)/3 |2,4

sooo im now at -4^2 + (4^3)/3- (2^2+ (2^3)/3)

which simplifies out to 48/3 + 64/3 + -12/3 -8/3 =92(pi)/3

this is obviously not the answer in the back of the book, but after picking my brain for HOURS i cant find my flaw :/
The attachment shows a sketch of this.
You are calculating the difference in areas of two concentric discs
and should be integrating over the y-direction from 2 to 4 using f(x)=y
and from 0 to 2 using fixed radii.

The outer disc has a constant radius of 4-1.