Area under curves

Question

Let $a$, $b$ > 0. Show that the area between the curves $y = x^a$ and $y = (1 - x)^b$ for 0 ≤ x ≤ 1 is equal to the area between $y = x^b$ and $y = (1 - x)^a. $

I created this demonstration, and it seems like this should hold true. I understand this can be done by integrating but I can't seem to proceed from there.

Answer 1

Hint

Take intuition from $$\int_0^1 f(x)dx=\int_0^1 f(1-x)dx$$

Answer 2

$$\text{Area} = \bigg|\int^1_0(x^a-(1-x)^b)dx\bigg| = \bigg|\int^1_0((1-x)^a-x^b)dx\bigg|$$

as, $$\int_0^af(x)dx = \int^a_0f(a-x)dx$$