How do you find the area between f(x)=(x-1)^3 and g(x)=x-1?
1 Answer
Explanation:
First, find the bounds given by
(x-1)^3 = x-1
(x-1)^3 - (x-1) = 0
(x-1)((x-1)^2-1) = 0
(x-1)(x^2-2x) = 0
x(x-1)(x-2) = 0
x = 0,1,2
This means the graphs actually close off 2 separate areas, since there are three intersection points. To see what I mean by this, here is a graph of
desmos.com/calculator
So, to find the total area, we need to find the area of both sections and then add them together.
From
int_0^1[(x-1)^3 - (x-1)]dx
int_0^1[x^3-3x^2+3x-1-x+1]dx
int_0^1(x^3-3x^2+2x)dx
[x^4/4-x^3+x^2]_0^1 = (1/4-1+1)-(0/4-0+0) = 1/4
From
int_1^2[(x-1)-(x-1)^3]dx
int_1^2[x-1-x^3+3x^2-3x+1]dx
int_1^2(-x^3+3x^2-2x)dx
[-x^4/4+x^3-x^2]_1^2 = (-16/4+8-4) - (-1/4+1-1) = 1/4
So the area of the first section is
Final Answer