Evaluation Of Definite Integrals By Substitution

To solve the problem, we are required to find the area enclosed between the line , the x-axis, and the vertical lines (ordinates) and using integration.

1. Understanding the Problem:
We are given the linear function:

We want to compute the area between this curve and the x-axis from to .

2. Identifying the Intersection with the X-axis:
Set to find the x-intercept:

This means the line cuts the x-axis at . So between and , the line is below the x-axis, and from to , the line is above the x-axis.

3. Splitting the Integral:
We split the total area into two parts and take the absolute value because area is always positive:

Between and , . Between and , .

4. Evaluating the First Integral:

5. Evaluating the Second Integral:

6. Total Area:
Add both areas:

Final Answer:
The required area is square units.