Question

How do you evaluate the definite integral by the limit definition given `intsqrt(r^2-x^2)dx` from [-r,r]?

Answer 1

This can be integrated geometrically.

Explanation:

The integrand `f(x) = sqrt(r^2-x^2)` has graph

`y = sqrt(r^2-x^2)`

which is the upper semicircle for the circle `x^2+y^2 = r^2`.

The integral from `-r` to `r` is the area of the semicircle.

`int_-r^r sqrt(r^2-x^2) dx = 1/2 pir^2`