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

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