With this question do I simply replace x in the integral, simplify and integrate?
"Using the substitution x=(a+b)/2 +((b-a)/2)cos t, evaluate Integral from a to b of square root of((x-a)(b-x))dx".
Do I need new limits?
Geometric interpretation: the integrand function is . Taking squares we see that represents a circle centered and radius . So the integral is half of the area of the corresponding disk, that is .
P.S. Prove It: surely you have a computation mistake, the integral must be positive.
It shouldn't make any difference if you substitute because , and when you make the substitution, it gets squared anyway, giving the same value.
You are correct that should integrate to , but that sign error doesn't make any difference since it goes to 0 from substituting both terminals anyway.
I have found what my mistake is. Since , that means when I took the square root, I should have done , not to get the positive value.