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

Answer 1

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.