Question

You are given the four points in the plane A=(-1,8), B=(2,-4), C=(6,4), and D=(10,-4). The graph of the function f(x) consists of the three line segments AB, BC and CD. Find the integral from -1 to 10 f(x)dx by interpreting the integral in terms of sums and/or differences of areas of elementary figures.

Answer 1

6

Explanation:

equation of AB:

`(y-8)/(-4-8) =(x-(-1))/(2-(-1))`

`3(y-8)=-12(x+1)`

`y=-4x+4`

When y = 0 → x = 1 → F = (1, 0)

equation of BC:

`(y-(-4))/(4-(-4)) =(x-2)/(6-2)`

`4(y+4)=8(x-2)`

`y=2x-8`

When y = 0 → x = 4 → G = (4, 0)

equation of CD:

`(y-4)/(-4-4) =(x-6)/(10-6)`

`4(y-4)=-8(x-6)`

`y=-2x+16`

When y = 0 → x = 8 → H = (8, 0)

integral f(x) from -1 to 10

= area ΔAEF - area ΔBFG + area ΔCGH - area ΔDHI

`=(1)/(2) (EF)(AE)-(1)/(2) (FG)(BB')+(1)/(2) (GH)(CC')-(1)/(2) (HI)(DI)`

`=(1)/(2) [(EF)(AE)-(FG)(BB')+(GH)(CC')-(HI)(DI)]`

`=(1)/(2) [(2)(8)-(3)(4)+(4)(4)-(2)(4)]`

`=6`