00:02
We need the volume of a shape that has this area here, right? and so then we have this function describing the area, right? so x equals r cosine theta, y equals r sine theta.
00:34
So then we have r cosine theta, theta squared plus r -sign theta squared equals 9.
00:46
So then r equals 3, right? and this goes from theta equals 0 to theta equals 3 pi over 2, right? now, yeah, we have the bounds, theta 0 to 3 pi over 2, right? and then zero, i mean, sorry, r equals 0 to r equals 3.
01:23
But also, we need to have the volume element of y, r, d, theta, okay? this is the area, this right here is the length.
01:48
Okay, but remember that y equals r sine theta, right? so then we have 0 to 3 pi over 2 times 0 to 3 of r sine theta r d d theta, right? so then that gives us 0 to 3 pi of 2 0 to 3 of r squared sine theta d theta and you can pull sine theta out of that inner integral and then you have 0 3 pi over 2 sine theta 0 to 3 of r squared dr d theta okay so you do that inner integral, alright? and then you get 0 to 3 pi over 2, sine, theta, i cubed over 3 from 0 to 3 d theta.
03:15
Okay? and this whole thing here is just 9.
03:21
Okay? so then you end up with 0 to 3 pi over 2, sine theta.
03:31
Times 9 times d theta...