Chapter 8.2, Problem 35E

To determine

To Find:

To find the area of the given regions by subtracting the area of a triangular region from the area of a larger region on or above the x-axis bounded by the curves $y^2=x+3$ and $y=2x$ .

Answer to Problem 35E

The area of the given regions bounded by the curves are $\approx 4.3333$ .

Explanation of Solution

Given information:

The given function is $y^2=x+3$ and $y=2x$ in the interval $0\le y\le \frac{\pi }{2}$ .

The red curve is the graph of $y^2=x+3$ and the blue line is the graph of $y=2x$ .

  

Now, we have to find the area by subtracting the triangle formed by $y=2x$ , $y=0$ and $x=1$ from the area formed by the entire region.

Consider $y^2=x+3\Rightarrow y=\sqrt{x+3}$ ,

So the value of y is both positive and negative. However, when we actually solve the equation, only the positive is used because we are trying to find the area above the x-axis.

Therefore, the area of the larger region is

  $A={\displaystyle \underset{-3}{\overset{1}{\int }}\sqrt{x+3}dx}$

Area of the triangle outside of area that we are trying to find but formed from larger region is

  $\frac{bh}{2}=\frac{1\left(2\right)}{2}$

Subtract the area of triangle from area of larger region,

  $A={\displaystyle \underset{-3}{\overset{1}{\int }}\sqrt{x+3}dx}-\frac{1\left(2\right)}{2}$

Substitute $u=x+3$ then $du=dx$ ,

When

  $\begin{array}{l}x=-3\Rightarrow u=0\\ x=1\Rightarrow u=4\end{array}$

  $\begin{array}{l}={\displaystyle \underset{0}{\overset{4}{\int }}\sqrt{u}du}-\frac{1\left(2\right)}{2}\\ ={\left[\frac{u^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+1}}{\frac{1}{2}+1}\right]}_0^4-\frac{1\left(2\right)}{2}\end{array}$

On Simplifying,

  $\begin{array}{l}={\left[\frac{2}{3}{u}^{\frac{3}{2}}\right]}_0^4-\frac{1\left(2\right)}{2}\\ =\frac{2}{3}\left[{\left(4\right)}^{\frac{3}{2}}-0\right]-\frac{1\left(2\right)}{2}\\ =\frac{16}{3}-\frac{1\left(2\right)}{2}\\ =\frac{16-3}{3}\\ =\frac{13}{3}\approx 4.3333\end{array}$

Therefore, the area of the given regions bounded by the curves are $\approx 4.3333$ .