Sketch the region bounded by the graphs of the given equations and find the area of their region.
$$x = y ^ { 2 } + 4 , x = y - 2 , y = - 2 , y = 2$$

Question

Quizplus · Accepted Answer

We can start by graphing the given equations:

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The region we are interested in is bounded by the lines $y=-2$ and $y=2$, and the parabola $x=y^2+4$ on top and the line $x=y-2$ on the bottom. We can see that the intersection point of the parabola and the line is $(3,1)$.

To find the area of this region, we need to integrate with respect to $y$ from $-2$ to $2$:

\begin{align*}
\int_{-2}^2 [(y^2+4)-(y-2)]dy &= \int_{-2}^2 (y^2-y+6)dy \
&= \left[\frac{1}{3}y^3-\frac{1}{2}y^2+6yight]_{-2}^2 \
&= \frac{32}{3}-8+12-\frac{8}{3} \
&= \frac{32}{3} \
\end{align*}

Therefore, the area of the region is $\frac{32}{3}$. The best choice is $\boxed{	extbf{(C)}}$.

Sketch the Region Bounded by the Graphs of the Given x=y2+4,x=y−2,y=−2,y=2x = y ^ { 2 } + 4 , x = y - 2 , y = - 2 , y = 2x=y2+4,x=y−2,y=−2,y=2

Sketch the Region Bounded by the Graphs of the Given $x = y ^ { 2 } + 4 , x = y - 2 , y = - 2 , y = 2$