Q. 43 - 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 $\frac{38}{6}$.

Step 1. Given information.

The given functions 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 function is,

Sep 3. Required Area.

The required exact area is,

$$\begin{gathered} {\int }_{-2}^2\left|f\right(x)-g(x\left)\right|dx={\int }_{-2}^{-1}\left(f\right(x)-g(x\left)\right)dx+{\int }_{-1}^2\left(g\right(x)-f(x\left)\right)dx \\ ={\int }_{-2}^{-1}\left(x^2-x-2\right)dx+{\int }_{-1}^2\left(x+2-x^2\right)dx \\ ={\left[\frac{x^3}{3}-\frac{x^2}{2}-2x\right]}_{-2}^{-1}+{\left[2x+\frac{x^2}{2}-\frac{x^3}{3}\right]}_{-1}^2 \\ =\frac{38}{6} \end{gathered}$$