How do you find the area between f(x)=(x-1)^3 and g(x)=x-1?

1 Answer
Jun 1, 2017

1/2

Explanation:

First, find the bounds given by f(x) = g(x)

(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 color(blue)(y=x-1 and color(red)(y = (x-1)^3:

desmos.com/calculatordesmos.com/calculator

So, to find the total area, we need to find the area of both sections and then add them together.

From x = 0 to x = 1, we can see that (x-1)^3 > x-1, so in order to find the POSITIVE area between the two curves, we will subtract x-1 (smaller) from (x-1)^3 (bigger).

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 x=1 to x=2, we can see that x-1 > (x-1)^3, so in order to find the POSITIVE area between the two curves, we will subtract (x-1)^3 (smaller) from x-1 (bigger).

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 1/4 and the area of the second section is 1/4. Therefore, the total area between f(x)=(x-1)^3 and g(x) = x-1 is 1/2

Final Answer