Prove that the integral of a 2lperiodic function is the same

Prove that the integral of a (2l)-periodic function is the same on any interval of length 2l. That is, let f(x) be (2l)-periodic, then prove that integral^l_-l f(x)dx = integral^a + l_a - l f(x)dx for any a R.

Solution

Both integrals are the areas of the same curve (y =f(x)) (because of the periodicity with period 2l)

over the same length (a+l-(a-l)) =2l.)

Hence the result