# Integration with Cylindrical shells

#### Fredrik373

Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

The question:
Find the volume of S using the method of cylindrical shells.

S is generated by rotating about the x-axis the region bounded by
y = x^2 , y = 0, and x = 1

(Nerd)

#### HallsofIvy

MHF Helper
Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

The question:
Find the volume of S using the method of cylindrical shells.

S is generated by rotating about the x-axis the region bounded by
y = x^2 , y = 0, and x = 1

(Nerd)
Since are rotating around the x-axis, and want to use "cylindical shells", you will want to integrate with resect to y.

The integral will be $$\displaystyle \int_0^1 2\pi r h dy$$ where "r" is the radius of the cylinder- that will be just "y", the distance from the x-axis to the line that, rotated around the x-axis, creates the cylinder, and "h" is the length of cylinder, the length of the line that creates the cylinder- that will be "1- x" where x is the x-coordinate of the point (x, y) at the left end of that line: since $$\displaystyle y= x^2$$, x= ?

• Fredrik373

#### skeeter

MHF Helper
Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

The question:
Find the volume of S using the method of cylindrical shells.

S is generated by rotating about the x-axis the region bounded by
y = x^2 , y = 0, and x = 1
shells w/r to y ...

$$\displaystyle V = 2\pi \int_0^1 y(1 - \sqrt{y}) \, dy$$

disks w/r to x ...

$$\displaystyle V = \pi \int_0^1 x^4 \, dx$$

• Fredrik373