Q. 41 - 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 $x=a to x=b.$ 

Step-by-Step Solution

Answer

The exact area of the region is$\frac{13}{2}.$

Step 1. Given Information.

The given functions and interval are $f\left(x\right)=1+x, g\left(x\right)=2-x,[0,3]$.

Step 2. Graph of the functions.

The graph of both the functions is ,

Step 3. Required Area.

The area will be,

$$\begin{gathered} {\int }_0^3\left|f\right(x)-g(x\left)\right|dx={\int }_0^{0.5}\left(g\right(x)-f(x\left)\right)dx+{\int }_{0.5}^3\left(f\right(x)-g(x\left)\right)dx \\ ={\int }_0^{0.5}(2-x-1-x)dx+{\int }_{0.5}^3(1+x-2+x)dx \\ ={\int }_0^{0.5}(1-2x)dx+{\int }_{0.5}^3(-1+2x)dx \\ =\left([x{]}_0^{0.5}-2{\left[\frac{x^2}{2}\right]}_0^{0.5}\right)+\left(-[x{]}_{0.5}^3+2{\left[\frac{x^2}{2}\right]}_{0.5}^3\right) \\ =\left[0.5-(0.5{)}^2\right]+\left[-(3-0.5)+\left(9-(0.5{)}^2\right)\right] \\ =6.5 \\ =\frac{13}{2} \end{gathered}$$