Q. 42 - 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$x=a to x=b.$

$f\left(x\right)=1+x, g\left(x\right)=x+2,[0,3]$

Step-by-Step Solution

Answer

The exact area is $-3.$

Step 1. Given Information.

The functions are $f\left(x\right)=1+x, g\left(x\right)=x+2$

The interval is $[0,3]$.

Step 2. Graph.

The graph of the functions is,

Step 3. Required area.

The area will be,

$$\begin{gathered} {\int }_0^3\left|f\right(x)-g(x\left)\right|dx={\int }_0^3\left(f\right(x)-g(x\left)\right)dx \\ ={\int }_0^3(1+x-x-2)dx \\ ={\int }_0^3(-1)dx \\ =-[x{]}_0^3 \\ =-3. \end{gathered}$$