How do you find the area between #f(x)=3(x^3-x)# and #g(x)=0#?

Answer 1

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Always draw a graph or sketch when finding an area so that you get an understanding of for the area in question:

# f(x)= 3(x^3-x) #

graph{ 3(x^3-x) [-2, 2, -3, 3]}

We can easily show that f(x) is an odd function:

# f(-x) = 3((-x)^3-(-x) ) # # " " = 3(-x^3+x ) # # " " = -3(x^3-x ) # # " " = -3f(x) #
The purpose of demonstrating this property is because now we can conclude that the area bound by the curve #f(x)= 3(x^3-x)# and #g(x)-=0# to the right of #Oy# is identical to that to the left of #Oy#. As one has positive contribution, and the other has negative contribution, then the net area is zero.

If however we wanted the positive contribution area only, then this would be given by:

# A = int_-1^0 \ 3(x^3-x) \ dx # # \ \ \ = 3 \ int_-1^0 \ x^3-x \ dx # # \ \ \ = 3 \ [x^4/4-x^2/2]_-1^0 # # \ \ \ = 3 \ {(0-0) - (1/4-1/2) }# # \ \ \ = 3 \ (1/4)# # \ \ \ = 3/4#
Similarly, the negative contribution would be #-3/4#

You should interpret these result depending upon the nature of the initial question!

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Answer 2

To find the area between ( f(x) = 3(x^3 - x) ) and ( g(x) = 0 ), you need to find the definite integral of ( f(x) ) from the points where it intersects the x-axis.

First, find the points where ( f(x) ) intersects the x-axis by setting ( f(x) = 0 ) and solving for ( x ). Then, calculate the definite integral of ( f(x) ) from the smallest x-intercept to the largest x-intercept.

The area between ( f(x) ) and ( g(x) ) is equal to the absolute value of the integral result.

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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.

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.

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