posted by jen .
You are given the four points in the plane a=(-2,2), b=(1,-3), c=(3,6), and d=(7,-3). The graph of the function consists of the three line segments AB, BC and CD. Find the integral by interpreting the integral in terms of sums and/or differences of areas of elementary figures
Here the integral is being evaluated, so the area of the elementary figures could be positive or negative.
It can be observed that all the line segments form a (twisted) trapezoid with the x-axis, i.e. part of the area is above the x-axis, and part of it is below.
Knowing that the area of a trapezoid is the average of the y-coordinate of the two ends, multiplied by the length, it will be possible to calculate the three areas separately.
For example, the area created by the segment AB is:
Average of y-coordinates = (2+(-3))/2=-0.5
Length = (1-(-2)) =3
Integral of segment AB=3*(-0.5) = -1.5
Repeat for the two other segments and add up the total.