Given a diagram from Calculus of a Single Variable by Larson and Edward (9th edition):
I am interested in finding the volume of various regions when rotated about various lines. Specifically, I am wondering if my set-up for finding the volume is correct; I have no issues simply integrating such functions. Note that this is not homework - I am just reviewing for an exam tomorrow.
Problem 1: $R_3$ about $x = 1$
I used a horizontal slice (disk), so my integral was $\pi\int_0^1 (1-\sqrt{y})^2 dy$ since every part of of the region $R_3$ is touching the axis of revolution.
Problem 2: $R_2$ about $x = 1$
Again, I used a horizontal slice, except this one was a washer. My integral was $\pi\int_0^1 (1-y)^2 - (1-\sqrt{y})^2 dy$
Just looking for a confirmation that this or correct or (if needed) an explanation of why I am wrong. Thanks.