find the volume of the prolate spheroid (a solid of revolution in the shape of a football ) obtained by revolving the region under the graph of the function y=3/5 √25-x^2 from x= -5 to x=5 about the x-axis...

volume of ellipsoid of semi-axes a,b,c is 4/3 pi * a*b*c

your ellipse is

5y = 3sqrt(25-x^2)
25y^2 = 9(25-x^2)
9x^2 + 25y^2 = 9*25
x^2/25 + y^2/9 = 1

so semi-axes are 5 and 3
rotate around the x-axis gives a third semi-axis of 3 (y-height)

volume is 4/3 * 5*3*3 pi = 60pi
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OK, if you insist on integration, use discs, where

v = Int(pi*r^2 dx)[-5,5]
where r = y
r^2 = 9/25 (25-x^2)

v = 2*9/25*pi*Int(25-x^2) dx [0,5]
= 18/25 pi * (25x - 1/3 x^3)[0,5]
= 18pi/25 * (125 - 125/3)
= 18pi/25 * 250/3
= 60pi