integral of (-5r to 5r) sqrt((3r^2-y^2)(25r^2-y^2))dy

Question

Integral from -5r to 5r square root of 3r^2-y^2)(25r^2-y^2 dy. 

Answer 1

I=∫_-5r^5r√(3r²-y²)(25r²-y²)dy=r²∫_-5⁵√(3-t²)(25-t²)dt.

This integral cannot be evaluated directly. It can be reduced (or expressed) to Elliptic Integrals.

Substitution y=5rsint; dy=5r cost dt, 

I=25r³∫√(3-25sin²t) cos²tdt=I=25r³∫√(3-25sin²t) (1-sin²tdt)=

=25r³∫√(3-25sin²t) dt +r³∫√(3-25sin²t)(-25sin²t) dt=

=25r³∫√(3-25sin²t) dt +r³∫√(3-25sin²t)(3-25sin²t-3)dt=

=22r³∫√(3-25sin²t) dt +r³∫√(3-25sin2t)³dt.