Q. 23 - Precalculus Enhanced with Graphing Utilities

Question

Use the result obtained in Problem $20$ to find the area enclosed by $f\left(x\right)=x^2+3x+5$, the $x-axis$ and the lines $x=-4$ and $x=4$.

Step-by-Step Solution

Answer

The area is $82.67$.

Step 1. Given information

It is given that $f\left(x\right)=x^2+3x+5$, we need to use the result obtained in Problem $20$ to determine the area enclosed by $f\left(x\right)$, the $x-axis$ and the lines $x=-4$ and $x=4$.

Step 2. Simplify

$f\left(x\right)=x^2+3x+5$

For $x=-4$,

$y_0={\left(4\right)}^2+3·(-4)+5$

     $=9$

For $x=0$,

$y_1={\left(0\right)}^2+3\left(0\right)+5$

     $=5$

For $x=4$,

$y_2={\left(4\right)}^2+3\left(4\right)+5$

    $=33$

So, Area $=\frac{2}{3}(y_0+4y_1+y_2)$

               $=\frac{2}{3}(9+4·5+33)$

               $=\frac{248}{3}$

               $=82.67$.