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# Integrals and Piecewise Functions

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The definite integral is the area between the function and the x-axis. If the function is below the axis, the area is negative.

We have a triangle between x=-2 and x=1 and height of 3, so the area of the function between -2 and 1 is:

Between x=1 and x=3 the area is a rectangle with height 3, thus the area in this section is:

The area between x=3 and x=5 is divided into two identical triangles, only one of them is below the axis (its area is negative). Hence the total area under the graph in this section is simply zero.

So the total area from x=-2 to x=5 is:

Then another integration by parts:

Thus:

And:

Integrating by parts:

But:

Hence:

Substitution:

Second substitution:

Therefore:

So:

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