Question

Consider the region between f and g on [0, 4] as in the

graph next at the left. (a) Draw the rectangles of the left-

sum approximation for the area of this region, with n = 8.

Then (b) express the area of the region with definite

integrals that do not involve absolute values.

Short answer

Part a:The graph is

Part b: The expression is

${\int }_0^1 \left(g\right(x)-f(x\left)\right)dx+{\int }_1^3 \left(f\right(x)-g(x\left)\right)dx+{\int }_3^4 \left(g\right(x)-f(x\left)\right)dx$

Step-by-step solution

Part  (a) Step 1. Given information 

Given a graph

Part (a) Step 2. To find the left sum approximation for the area of shaded region .

Part (b) Step 1: To represent the area using definite integral . 

${\int }_0^1 \left(g\right(x)-f(x\left)\right)dx+{\int }_1^3 \left(f\right(x)-g(x\left)\right)dx+{\int }_3^4 \left(g\right(x)-f(x\left)\right)dx$

Therefore the area is ,${\int }_0^1 \left(g\right(x)-f(x\left)\right)dx+{\int }_1^3 \left(f\right(x)-g(x\left)\right)dx+{\int }_3^4 \left(g\right(x)-f(x\left)\right)dx$