Chapter 7.2, Problem 12E

To determine

To find: The area enclosed by the graph.

Answer to Problem 12E

The area is: $A=\frac{343}{48}$

Explanation of Solution

Given information:

The functions are:

  $\begin{array}{c}{y}^2=x+3,\\ y=2x\end{array}$

Calculation:

The graph is shown below:

  

Solve each equation for $x$ :

  $\begin{array}{c}x=y^2-3\\ \Rightarrow x=\frac{y}{2}\end{array}$

Find the point at which the two equations meet.

  $\begin{array}{c}{y}^2-3=\frac{y}{2}\\ \Rightarrow 2y^2-y-6=0\\ \Rightarrow (2y+3)(y-2)=0\\ \Rightarrow y=-\frac{3}{2},2\end{array}$

The integral of the right curve less the left curve determines the space between the curves.

  

  $\begin{array}{c}\Rightarrow A={\left[\frac{y^2}{4}-\frac{1}{3}{y}^3+3y\right]}_{-3/2}^2\\ \Rightarrow A=1-\frac{8}{3}+6-\left(\frac{9}{16}+\frac{9}{8}-\frac{9}{2}\right)\\ =\frac{343}{48}\end{array}$

Therefore, the required area of the shaded region analytically is $\frac{343}{48}$ .