Question

As viewed from above, a swimming pool has the shape of the ellipse x24900+y22500=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. v = cubic feet

Answer 1

4900>2500, the major axis of the ellipse is along x-axis
a=sqrt(4900)=70
b=sqrt2500=50

The squares are parallel to the y-axis, thus their sides extend in the y-axis direction, hence the length of the sides are:
y=+/-50sqrt(1-x^2/4900))
The area of the square
=y^2=[2500-2500(x^2/4900)

Thus, the volume will be sum along the x-axis from one of the ends of the ellipse to the other. by definition, the limits are (-a to a)=(-70 to 70)
But since the ellipse is symmetrical we can go from (0 to 70) and double the integral.
thus
V=2*int(0 to 70)[2500-2500(x^2/4900)]dx
=int(0 to 70)[5000-(50/49)x^2]dx
=[5000x-50/147x^3] (0 to 70)
=233,333 1/3 ft^3