Integration with piecewise defined function and jump discontinuity

Question

I have a math problem:

$\int^\limits\pi_0f(x)dx$

where

$f(x) = \sin x,\mbox { if } 0\leq x<\pi/2$

$f(x) = \cos x, \mbox { if } \pi/2\leq x \leq \pi$

My problem is that there is a jump discontinuity at $x = \pi/2$

I saw that people evaluate the integral for each function of the piecewise defined function and then add them up but I feel its a bit wrong because, they explain it as drawing a rectangle to the point of the open circle in the graph and my question is how can that be since domain of $\sin x$ in this function does not include $\pi/2$

I know my grammar is quite bad, but I hope I am getting my message across. Can someone please give me clarity on this?

Answer 1

The intuition, without delving into the theory, for why this is valid is that the "width" of a point is zero so the amount this point contributes to the integral is zero.