Find the Volume of water in pool using double integrals

[mridgwa]

Homework Statement

The depth of water in a swimming pool fits the equation f(x,y) = 2sin (x/20 - 7) - 3 cos ( x-3 /5)+8 when 0<=x<=20 and the sides of the pool fir the equations y(x) = 10-(x-10)^2/10 and y(6)= (x-10)^2/20 -5

Find the volume of the water in the pool using a double integral

I am not sure how to set up this integral. Any help in setting it up will be greatly appreciated.

 

[HallsofIvy]

Start by graphing the two sides- they will be parabolas, of course. Imagine a thin rectangle perpendicular to the x-axis. The length of that rectangle will be the difference of the two y values for the side-that will, of course,depend upon x. The depth is also a function of x only so you don't really need a double integral. Just integrate the product of those two lengths (the area of the rectangle) with respect to x.

If you really do want to use a double integral, get that "length" of the rectangle by integrating $\int dy$ with the the parabolic functions as limits.