Mathematics: As viewed from above, a swimming pool has the shape of the ellipse x21600+y2400=1, where x and y are measured in feet. the cross sections perpendicular to the x-axis are squares. find the total volume of the pool.

As viewed from above, a swimming pool has the shape

inucornspineapple

09.01.2022

I suppose the ellipse should have the equation

$\dfrac{x^2}{1600}+\dfrac{y^2}{400}=1\implies y=\pm\dfrac{\sqrt{1600-x^2}}4$

Then the cross sections are squares with side lengths

$\dfrac{\sqrt{1600-x^2}}4-\left(-\dfrac{\sqrt{1600-x^2}}4\right)=\dfrac{\sqrt{1600-x^2}}2$

so that each section contributes an area of

$\left(\dfrac{\sqrt{1600-x^2}}2\right)^2=\dfrac{1600-x^2}4$

Then the total volume of the pool would be given by

$\displaystyle\int_{-40}^{40}\frac{1600-x^2}4\,\mathrm dx=\boxed{\frac{64,000}3}$

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