Chapter B, Problem 53E

To determine

To calculate: To sketch the region bounded by the curves

Answer to Problem 53E

The region bounded by the curves is sketched

Explanation of Solution

Given information: Equation of curves are $x+4y=8$ and $x=2y^2-8$

Calculation:

Equation of curves are given as follows:

  $x+4y=8$ --------(1)

  $x=2y^2-8$ -------(2)

Substituting the value of $x=2y^2-8$ in equation (1)

  $\begin{array}{l}2y^2-8+4y=8\\ 2y^2+4y-16=0\\ y^2+2y-8=0\\ y^2+4y-2y-8=0\\ y\left(y+4\right)-2\left(y+8\right)=0\\ \left(y+4\right)\left(y-2\right)=0\\ y=-4,2\end{array}$

Now, substituting the values of $y=-4$ in equation (1)

  $\begin{array}{l}x+4y=8\\ x+4\left(-4\right)=8\\ x=8+16\\ x=24\end{array}$

Thus, one point of intersection is $\left(24,-4\right)$

Again, substituting the values of $y=2$ in equation (1)

  $\begin{array}{l}x+4y=8\\ x+4\left(2\right)=8\\ x=8-8\\ x=0\end{array}$

Thus, one point of intersection is $\left(0,2\right)$

Hence the area between the curves is displayed as follows:

  

Conclusion:

Hence, region bounded by the curves is sketched