Single Variable Calculus, Chapter 6, 6.1, Section 6.1, Problem 4

Determine the area of the shaded region



$
\begin{equation}
\begin{aligned}
A &= \int^{y_2}_{y_1} (x_{\text{right}} - x_{\text{left}}) dy\\
\\
A &= \int^3_0 \left[\left( 2y - y^2\right) - \left( y^2 - 4y \right) \right] dy\\
\\
A &= \int^3_0 \left( -2y^2 + 6y \right) dy\\
\\
A &= \left[ \frac{-2y^3}{3} + \frac{6y^2}{2} \right]^3_0\\
\\
A &= \left[ \frac{-2(3)^3}{3} + \frac{6}{2} (3)^2 \right] - \left[ \frac{-2(0)^3}{3} + \frac{6(0)^2}{2} \right]\\
\\
A &= 9 \text{ units}^2
\end{aligned}
\end{equation}
$