1
B
1
1
A 3
We can therefore interpret the integral as the area under the region from -1 to 3, which is represented by the two rectangles. Thus, Area under region = Sum of areas of triangles A and B. Area of A
=
½ bh
=
½(3 × 3)
=
½(9)
=
4.5
Area of B
=
½ bh
=
½(1 × -1)
=
½(-1)
=
-0.5
Therefore, Total Area
=
Area of A + Area of B
=
4.5 + (-0.5)
=
4.0
Thus, the area under the line y = 2 – x from -1 to 3 is 4.
Example 8 Evaluate the integral by interpreting it in terms of area.
∫
2 -2
( 1 – |x|) dx
SOLUTION We start by graphing the function f(x) = 1 - |x| on the interval [-2, 2]. (See the next page). From the graph, we see that the integral can be interpreted as the sum of the areas of the triangles A, B and C.