- #1
mr3369
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Homework Statement
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.
Homework Equations
the function is y = -x^2+6x -8
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.