Chapter 8.2, Problem 14E

To determine

To find the total shaded area .

Answer to Problem 14E

The total shaded area is $\frac{49}{6}$ squared units.

Explanation of Solution

Concept used: Area between curves.

If f and g are continuous with $f\left(x\right)\ge g\left(x\right)$ throughout [a,b], then the area between

The curves $y=f\left(x\right)\text{ and }y=g\left(x\right)$ from a to b is the integral of $\left[f-g\right]$ from ato b.

  $\begin{array}{l}A={\displaystyle {\int }_a^b\left[f\left(x\right)-g\left(x\right)\right]}dx\\ \end{array}$

Given information:

The given equations is, $y=4-x^2\text{ , }y=-x+2$ .

  

Calculation: The total shaded area is obtained as,

The curves given by, $y=-x+2$ and $y=4-x^2$ intersect when $x=-1\text{ and }x=2.$

Using these fact,We can split the integration up into 3 parts.

Now, area bounded by these region is:

  $\begin{array}{l}\text{A}={\displaystyle {\int }_a^b\left[f\left(t\right)-g\left(t\right)\right]}dt\\ ={\displaystyle {\int }_{-2}^{-1}\left(-x+2\right)-\left(4+x^2\right)dx}+{\displaystyle {\int }_{-1}^2\left(4-x^2\right)-\left(x+2\right)dx}+{\displaystyle {\int }_2^3\left(-x+2\right)-\left(4-x^2\right)dx}\\ =\frac{49}{6}\end{array}$

Hence, the total shaded area is $\frac{49}{6}$ squared units