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

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

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*

*Sign up to view the whole answer*

*By signing up, you agree to our Terms of Service and Privacy Policy*

*Sign up with email*

*Answer 2Sign up to view the whole answerSign up with email*

To find the area between the curves (f(x) = (x-1)^3) and (g(x) = x - 1), first, determine the points of intersection by setting (f(x) = g(x)) and solving for (x). Then, integrate the absolute difference between the two functions over the interval where they intersect. The integral will be (\int_{a}^{b} |f(x) - g(x)| , dx), where (a) and (b) are the x-coordinates of the points of intersection. Calculate the integral to find the area.

By signing up, you agree to our Terms of Service and Privacy Policy

*Answer from HIX Tutor*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

*Trending questions*

*If #y'(x)+y(x)g'(x)=g(x)g'(x)" ; y(0)=0 ; g(0)=g(2)=0 " where " x in RR# then #y(2)=?#**The region under the curve #y=1/x# bounded by #1<=x<=2# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?**Let R be the region in the first quadrant enclosed by the hyperbola #x^2 -y^2= 9#, the x-axis , the line x=5, how do you find the volume of the solid generated by revolving R about the x-axis?**How do you find the integral #int_0^1x*e^(-x^2)dx# ?**How do you find the volume of a solid that is enclosed by #y=sqrt(4+x)#, x=0 and y=0 revolved about the x axis?*

*Not the question you need?*

*HIX TutorSolve ANY homework problem with a smart AI*

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7