Question

Find the area of the shaped region.

Short answer

The area of the shaded region is\(9.\)

Step-by-step solution

Given information

Consider the functions \(f(y)\)and \(g(y)\)such that \(f(y) \ge g(y)\)for \(y \in (c,d).\)

The area bounded between \(x = f(y),x = g(y)\) and the horizontal lines \(y = c,d\) is given by \(A = \int_c^d f (y) - g(y)dy\)

Determining the area of shaped region

From the given graph, we can see that \(2y - {y^2} \ge {y^2} - 4y\)on the Interval \(y \in (0,3)\)

Therefore, the shaded area is

\(\begin{aligned}\int\limits_0^3{\left({\left({2y -{y^2}}\right)-\left( {{y^2} -4y}\right)} \right)dy}\\\int_0^3 2 y -{y^2} -{y^2} + 4ydy\\\int_0^3 6y-2{y^2}dy\end{aligned}\)

\(\begin{aligned} &= \left[ {\frac{{6{y^{1 + 1}}}}{{1+1}}- \frac{{2{y^{2+ 1}}}}{{2 + 1}}} \right]_0^3\\ &= \left[ {3{y^2} -\frac{{2{y^3}}}{3}} \right]_0^3\\ &=\left[ {3 \cdot {3^2} - \frac{{2 \cdot {3^3}}}{3}} \right] - \left[ {3 \cdot {0^2} - \frac{{2 \cdot {0^3}}}{3}} \right]\\ =& \left[ {3 \cdot {3^2} - 2 \cdot {3^2}} \right] - 0\\ &= (3 - 2) \cdot {3^2}\\ &={3^2}=9\\A&=9\end{aligned}\)