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= ?