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This all stems from me approximating pi by numerically evaluating the integral S f(x)dx from 0 to 1 and multiply the sum by 4.

Now...

Would you agree that f(x) has a derivative

f'(x) = (1 - x^2)^-1/2 * -2x

?

According to my textbook this is so. Now I can easily find a primary function for f(x).

F(x) = (1 - x^2)^3/2 / -2x

Now it doesn't seem possible to evaluate [ F(x) ] from 0 to 1.

Though it should yeild pi/4, it doesn't.

Doing a riemann sum produces an approximation to pi, while evaluating [ F(x) ] only returns bogus. Since pi is an irrational number I accept that it is impossible to express it exactly. Though, I would like someone to explain why this doesn't work.