Thread: Rotational volume problem

Hi everyone!

I am having a bit trouble with a question about rotational volume. So, this is the problem:

"An area R in the xy plane is given by:

0 ≤ x ≤ 1

and

0 ≤ y ≤ x³

Find the volume when rotating R around the axis y = -1"

Here is how I was thinking. First, I drew this figure:




My idea was to draw the boundaries, and two radius in red, which I need to find in order to use the washer method. So, I thought the lower boundary must be r(y) = g(y) - c = x³ - (-1) = x³ + 1 (The second red line in the picture)

But I am unsure what I should put as the upper boundary (The first radius in the picture).

I hope someone can help and tell me in what way I am thinking wrong.

Thanks!

The region is bounded by .

An arbitrary slice is the washer:



The outer radius is:



The inner radius is:



The "stack" of washers runs from to , so summing the washers in the stack, we find the volume is:



Expand and simplify the integrand, then use the anti-derivative form of the FTOC to compute the volume.

Thank you so much for the help, I understood it much better now, I also noticed my figure is completely wrong.... How embarrassing