you are given the four points in the plane A=(-2,-2) , B=(1,7),C=(4-5) and D=(7,1). The graph of the function f(x) consists of the three line segments AB,BC,CD.

Question

Find the integral following integral by interpreting it in terms of sums and/or differences of areas of elementary figures from drawing graph f(x). Find the integral from -2 to 7 f(x) dx .

Answer 1

The shapes between the points are trapezoids. Each y-value is a base and the height is the distance between the x-vaules. Some of the areas are negative and this is okay. In your picture you can add additional details like x-intercepts and find the area of those trapezoids. In terms of the area you can just use 3 trapezoids. Using the area of a trapezoid as A = 1/2(b₁+b₂)h,

A = 1/2(-2+7)3 + 1/2(7-5)3 + 1/2(-5+1)3 = 9/2.