1. The problem statement, all variables and given/known data the function is y = -x^2+6x -8 suppose a city is surrounded by a ring of mountains and these mountains can be illustrated by rotating the above function around the y-axis. Find the volume of the earth that makes up these mountains. Suppose the city suffers from air pollution and wants to build a system to reduce this air pollution. However, in order to do so, it needs to find the volume of polluted air. This volume is the region between the y-axis and the function, rotated about either x or y axis. The picture for the polluted air is attached. 2. Relevant equations the function is y = -x^2+6x -8 3. The attempt at a solution For the first part of finding the volume of earth, I integrated the function, using the cylindrical shell method. And I got the answer as 8 pi. I used the radius as r = x, height as h = -x^2+6x -8, So I integrated 2pi*r*h, from 2 to 4 which are the x intercepts. For the 2nd part, I made the function in terms of y, So first I did completing the square and got y=-(x-3)^2 +1 So that gives me x = sqrt(1-y) +3 I used x as the radius, and using disk method , revolved the function around the y-axis. So radius = sqrt(1-y) +3 , Then i integrated pi(r^2) from 0 to 3. However, my answer gives me root(-2) which isn't possible. So I am stuck in this 2nd part.