When we look at the graph, we see a big triangle below the x-axis and a little triangle above the x-axis.

The total area above the x-axis and below f is

The total area below the x-axis and above f is

To get the integral of f, we take the area above the x-axis and subtract the area below the x-axis:

Example 2

Find

This graph consists of two triangles, both below the x-axis:

The total area between the x axis and -|x| above the axis is 0. The total area between the x-axis and -|x| below the axis is 20. So

Example 3

If , what is a?

Draw a picture. We're integrating 2x from a to 1.

Since the integral value is negative there must be some area below the x-axis, so we can assume a is negative.

This line forms two triangles. We know the dimensions of the one above the x-axis:

The area of the triangle above the x-axis is 1.

For the triangle below the x-axis, we can say what its dimensions are in terms of a. Since a is negative, the base of the triangle is the positive distance -a. Similarly, the height of the triangle is -2a.

The area of this triangle is

We now have

This is a perfectly reasonable equation that we can solve for a.

Since we know a needs to be negative, a = -3 is the answer.