# Thread: Finding the Volume of a Pool

1. ## Finding the Volume of a Pool

Find the Total Volume of a Pool:

As viewed from above, a swimming pool has the shape of the ellipse

x^2/900+y^2/400 = 1

The cross sections perpendicular to the ground and parallel to the y-axis are squares. Find the total volume of the pool. (Assume the units of length and area are feet and square feet respectively. Do not put units in your answer.)
V =

how do I do this problem?

I've tried several different formulations, and I figured the answer was Int (subscript -30, superscript 30) (X^2)-900 dx but this does not appear to be the correct answer.

2. If we take our pool equation and solve for y we get:

$y=\frac{2}{3}\sqrt{900-x^{2}}$

This is the length of the base of the square as the cross sections stack up along the x-axis.

But the cross-sections are squares:

$\frac{16}{9}\int_{0}^{30}\left(\sqrt{900-x^{2}}\right)^{2}dx$

We squared our function and multiplied by 4 because the integral gives the area in one quadrant of the ellipse.

What we end up with is a very easy integral once it is set up.

3. That would seem to get me the final answer of 32000, which does not seem to work.

Am I calculating incorrectly?

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### (x y)^2=900

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