# Change of variable

Moving the red dot on the left with the mouse pointer selects a region under the curve $$y = \sin^2(x)\cos(x)$$. By the change of variable formula, the area of the region in blue on the left equals the area of the region in green under the curve $$y = u^2$$ on the right. That is, with the change of variable $$u = \sin(x)$$, $\int_0^a \sin^2(x)\cos(x)dx = \int_0^{\sin(a)} u^2du.$