Q. 44 - Calculus

Question

For each pair of functions $f$ and $g$ and interval $[a,b]$ in Exercises 41–52, use definite integrals and the Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f$ and $g$ from $x=a and x=b.$.

$f\left(x\right)=x^2, g\left(x\right)=x-2, [-2,2]$

Step-by-Step Solution

Answer

The exact area is $-\frac{4}{3}.$

Step 1. Given Information.

The given function and interval is $f\left(x\right)=x^2, g\left(x\right)=x-2, [-2,2]$.

Step 2. Graph of the functions.

The graph of the functions is,

Step 3. Required area.

The exact area will be,

$$\begin{gathered} {\int }_{-2}^2\left|f\right(x)-g(x\left)\right|dx={\int }_{-2}^2\left(f\right(x)-g(x\left)\right)dx \\ \mathbf{=}{\mathbf{\int }}_{\mathbf{-}\mathbf{2}}^{\mathbf{2}}\left(x^2-x-2\right)\mathit{d}\mathit{x} \\ ={\left[\frac{x^3}{3}-\frac{x^2}{2}-2x\right]}_{-2}^2 \\ =\left[\frac{2^3}{3}-\frac{2^2}{2}-2\left(2\right)\right]-\left[\frac{(-2{)}^3}{3}-\frac{(-2{)}^2}{2}-2(-2)\right] \\ =\frac{16}{3}-8 \\ =-\frac{4}{3} \end{gathered}$$