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 \(u = \sin(x)\), \[ \int_0^a \sin^2(x)\cos(x)dx = \int_0^{\sin(a)} u^2du. \]

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