Volumes by Cylindrical Shells: y=x3​, y=0, x= 1. x=2; about y-axis

[TheEpicnor]

Hi all!
I don't know how to do this problem, could I get some help please? It would be greatly appreciated!

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the following curves about the specified line.

(a) y=x³​, y=0, x= 1. x=2; about the y-axis

 

[Steven G]

Hi all!
I don't know how to do this problem, could I get some help please? It would be greatly appreciated!

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the following curves about the specified line.

(a) y=x³​, y=0, x= 1. x=2; about the y-axis

Can you please show us your graph for this problem and please label everything. Then please let us know where you are having trouble. Thank you.

 

[HallsofIvy]

At a given value of x, from 1 to 2, a vertical line will go from y= 0 up to \(\displaystyle y= x^3\) so of length \(\displaystyle x^3\). Rotating that around the y-axis, it sweeps out a cylinder with radius x and \(\displaystyle x^3\) so with surface area \(\displaystyle (2 pi x)(x^3)= 2\pi x^4\). Taking the thickness to be "dx" the total volume is \(\displaystyle 2\pi\int_1^2 x^4 dx\).