Question

Use the result obtained in Problem $20$to find the area enclosed by $f\left(x\right)=-5x^2+8$the $x-axis$and the lines$x=-1$and$x=1$

Short answer

The area is $12.67$.

Step-by-step solution

Step 1. Given information

It is given that $f\left(x\right)=-5x^2+8$, we need to use the result of the problem $20$to determine the area enclosed by$f\left(x\right)$, the$x-axis$and the lines$x=-1$and$x=1$.

Step 2. Simplify

$f\left(x\right)=-5x^2+8$

For, $x=-1$,

$f(-1)=-5{(-1)}^2+8$

$=3=y_0$

For $x=0$,

$f\left(0\right)=-5{\left(0\right)}^2+8$

$=8=y_1$

For $x=1$,

$y=-5{\left(1\right)}^2+8$

$=3$$=y_2$

The area is,

Area $=\frac{1}{3}(y_0+4y_1+y_2)$

$=\frac{1}{3}(3+4·8+3)$

$=\frac{38}{3}$

$=12.67$

$=\frac{38}{3}$