Problem

Evaluate

Solution

We can either use Riemann sums or look at the graph of the function to find the value of the integral. With a simple function like this, it's almost always easiest to see what the function looks like rather than trying to take limits of Riemann sums.

Call region A (in blue) the region under the graph from x = –3 to x = 3, and region B (magenta) the region under the graph from x = 3 to x = 6. Treating definite integrals as areas under curves, we know that the total integral which we're trying to find is simply the area of region A + the area of region B.

First, let's find region A's area. Since f(x) is an even function, we know

so to find the total area of region A, we can find the area under the graph from x = 0 to x = 3 and double this value. This is the area of a triangle with base = 3 and height 3, so its area is

The area of region A is twice this,

Now for region B. This is the area under the graph from x = 3 to x = 6. We see this is equal to the area of a rectangle plus the area of a triangle. These values are easy to calculate, simply by looking at the graph.

Rectangle's area:

Triangle's area:

Area of region B:

So the total area, and the value of the integral, is