What is the net area between #f(x) = e^(2x)-xe^x # and the x-axis over #x in [2, 4 ]#?

Answer 1

#=1/2e^8-7/2e^4+e^2#

We can split the integral up:

#int_2^4 (e^(2x) - xe^x)dx = int_2^4 e^(2x)dx - int_2^4 xe^xdx#

The first integral is easy

#[1/2e^(2x)]_2^4 = 1/2(e^8-e^4)#

For the second, we need to use integration by parts:

#int u*v' = u*v - int u'*v#
Let #u(x) = x# and #v'(x) = e^x# then #u'(x) = 1# and #v(x) = e^x#
#int xe^xdx = xe^x - int e^xdx = e^x(x-1)#

Therefore, we have

#1/2(e^8-e^4) - [e^x(x-1)]_2^4#
#1/2(e^8-e^4) -[3e^4 - e^2]#
#=1/2e^8-7/2e^4+e^2#
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Answer 2

To find the net area between ( f(x) = e^{2x} - xe^x ) and the x-axis over ( x ) in the interval ([2, 4]), we integrate ( f(x) ) with respect to ( x ) over the given interval and take the absolute value of the result.

The integral of ( f(x) ) over ([2, 4]) is:

[ \int_{2}^{4} (e^{2x} - xe^x) , dx ]

Using integration techniques, this can be calculated as follows:

[ \int_{2}^{4} (e^{2x} - xe^x) , dx = \left[\frac{e^{2x}}{2} - \frac{x^2e^x}{2} - \frac{xe^x}{2}\right]_{2}^{4} ]

Substituting the upper and lower limits:

[ \left[\frac{e^{8}}{2} - \frac{16e^4}{2} - 4e^4\right] - \left[\frac{e^{4}}{2} - \frac{4e^2}{2} - 2e^2\right] ]

[ = \left(\frac{e^{8}}{2} - 10e^4\right) - \left(\frac{e^{4}}{2} - 3e^2\right) ]

[ = \frac{e^{8}}{2} - 10e^4 - \frac{e^{4}}{2} + 3e^2 ]

[ = \frac{e^{8}}{2} - \frac{e^{4}}{2} - 10e^4 + 3e^2 ]

This gives us the net area between ( f(x) ) and the x-axis over ( x ) in the interval ([2, 4]).

Calculating the numerical value would require evaluating these exponential functions, which can be done using a calculator or computer software.

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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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