# Calculus

Consider the solid obtained by rotating the region bounded by the following curves about the line x=1.
y=x,y=0,x=4,x=6
Find the volume
So it would be
pi (integral from 3 to 6) of ((1-y)^2 -(1-0)^2) right?
so then you integrate it and get
pi(Y^3/3-y^2) from 3 to 6.
?

1. 👍 0
2. 👎 0
3. 👁 139
1. If you are going to integrate over y, the solid has two parts: a plain old cylinder of height 4 and thickness 2, and a variable-thickness shape of height 2.

So,

v = π(5^2-3^2)(4) + ∫[4,6] π(R^2-r^2) dy
where R=5 and r=x-1=y-1
v = 64π + π∫[4,6] 25-(y-1)^2 dy
= 64π + π∫[4,6] -y^2+2y+24 dy
= 64π + 52/3 π
= 244/3 π

I think shells are easier in this case. SO, since each shell has thickness dx, we have

v = ∫[4,6] 2πrh dx
where r = x-1 and h = y = x
v = 2π∫[4,6] (x-1)(x) dx
= 2π∫[4,6] x^2-x dx
= 244/3 π

1. 👍 0
2. 👎 0
posted by Steve

## Similar Questions

1. ### calculus

Sketch the region bounded by the curves y = x^2, y = x^4. 1) Find the area of the region enclosed by the two curves; 2) Find the volume of the solid obtained by rotating the above region about the x-axis; 3) Find the volume of the

asked by Megan on June 9, 2011
2. ### calculus

Consider the solid obtained by rotating the region bounded by the given curves about the line x = 4. x= 4y^2 , x = 4 Find the volume V of this solid.

asked by JESSICA on February 17, 2010
3. ### CALCULUS 2

Consider the solid obtained by rotating the region bounded by the given curves about the line x = 1. y = x, y = 0, x = 4, x = 6 Find the volume V of this solid

4. ### calculus

Consider the solid obtained by rotating the region bounded by the given curves about the line x = 6. y = sqrt(x), y = x Find the volume V of this solid.

asked by bob on January 31, 2009
5. ### Calculus Help Please Urgent!!!

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 5x^2, y = 5x, x ≥ 0; about the x-axis V = ??? Sketch the region

asked by Top on May 3, 2014
6. ### calculus

Find the volume of the solid obtained by rotating the region bounded by the curves y = x^8, y = 1 about the line y = 5.

asked by maggie on March 2, 2017
7. ### Calculus ll - Volumes

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=e^(-x), y=1, x=2; about y=2.

asked by Abby on October 7, 2010
8. ### K

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 3e^(-x), y = 3, x = 2; about y = 6

asked by Gagan on February 24, 2017
9. ### calculus

Find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=-3 y=x^2, x=y^2

asked by Anonymous on December 15, 2010
10. ### calculus 2

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x, y = 0, x = 2, x = 6; about x = 1

asked by TayB on January 27, 2016
11. ### calculus

Find the volume of the solid obtained by rotating the region bounded by the curves y=x^6, y=1 about the line y=4

asked by Anonymous on December 14, 2010

More Similar Questions