Q. 52 - Calculus

Question

For each pair of functions $f,g$ 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$x=a to x=b$

$f\left(x\right)=\frac{3}{x-2}, g\left(x\right)=6-x, [a,b]=[3,7]$

Step-by-Step Solution

Answer

The exact area is $2.237 .$

Step 1. Given Information.

The given function and interval is$f\left(x\right)=\frac{3}{x-2}, g\left(x\right)=6-x, [a,b]=[3,7].$

Step 2. Graph of the function.

The graph of the functions is,

Step 3. Required area.

The exact area will be,

$$\begin{gathered} {\int }_3^7\left|f\right(x)-g(x\left)\right|dx={\int }_3^5\left(g\right(x)-f(x\left)\right)dx+{\int }_5^7\left(f\right(x)-g(x\left)\right)dx \\ ={\int }_3^5\left(6-x-\frac{3}{x-2}\right)dx+{\int }_5^7\left(\frac{3}{x-2}-6+x\right)dx \\ ={\left[-\frac{x^2}{2}+6x-3\ln (x-2)\right]}_3^5+{\left[3\ln (x-2)+\frac{x^2}{2}-6x\right]}_5^7 \\ =-3\ln \left(3\right)+4-3\ln \left(3\right)+3\ln \left(5\right) \\ \approx 2.237 \end{gathered}$$