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What is the net area between #f(x)=x^3-x^2+5# in #x in[2,5] # and the x-axis?

Answer 1

Aurora Alvarado

128.25

the area under a curve from point a to point b is the Definite integral of that curve evaluated from point a to point b

therefore, area from a to b under the curve #f(x) = int_a^b f(x) dx #

therefore, after integrating your above function we get its antiderivative as #F(x) = 1/4x^3 - 1/3x^2 + 5x #

the antiderivative of #f(x)# Is usually named as #F(x)#

now, Subtract the value of #F(x)# at #a# from its value at #b#

and you get #128.25# =

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Answer 2

Christopher Agresta

To find the net area between ( f(x) = x^3 - x^2 + 5 ) and the x-axis on the interval ([2, 5]), you need to integrate the absolute value of the function over that interval. This is because the graph may lie above and below the x-axis, and we're interested in the total area between the curve and the x-axis.

The integral can be set up as follows:

[ \text{Net Area} = \int_{2}^{5} |x^3 - x^2 + 5| , dx ]

You would then evaluate this definite integral using the properties of absolute value functions.

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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.