   Chapter 4.2, Problem 35E

Chapter
Section
Textbook Problem

# Evaluate the integral by interpreting it in terms of areas. ∫ − 1 2 ( 1 − x ) d x

To determine

To evaluate:

The integral -12(1-x)dx by interpreting it in terms of areas.

Explanation

1) Concept:

A definite integral can be interpreted as a net area, that is, as difference of areas;

abf(x)dx=A1-A2

where A1 is the region above x- axis and below the graph of , and A2 is the region below x- axis and above the graph of f.

2) Formula for Area of a triangle:

A=12bh

where b is the base and h is the height

3) Given:

-12(1-x)dx

4) Calculation:

-12(1-x)dx  this given integral can be interpreted as the area under the graph of

fx= 1-x  between x= -1 and x=2

The graph of y= 1-x  is the line with the slope -1  as shown in the figure below.

Now, find the integral as the difference of the areas of the two triangles

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