Q. 36 - Calculus

Question

For each function f and interval [a, b] in Exercises 26–37, use definite integrals and the Fundamental Theorem of Calculus to find the exact values of (a) the signed area and (b) the absolute area of the region between the graph of f and the x-axis from x = a to x = b.

$f\left(x\right)=\frac{3-2x}{x^2},[a,b]=[1,4]$

Step-by-Step Solution

Answer

Area=$\frac{2}{9}$

Step 1. Given information

$f\left(x\right)=\frac{3-2x}{x^2},[a,b]=[1,4]$

Step 2. Finding the signed area

$\begin{array}{r}{\int }_1^3 \frac{1}{2x^3}dx\\ ={\left[-\frac{1}{4x^2}\right]}_1^3\\ =\frac{-1}{4(3{)}^2}+\frac{1}{4\left(1\right)}\\ =\frac{-1}{36}+\frac{1}{4}\\ =\frac{-1+9}{36}\\ =\frac{8}{36}\\ =\frac{2}{9}\end{array}$