What is the net area between #f(x) = e^(2x)-xe^x # and the x-axis over #x in [2, 4 ]#?
We can split the integral up:
The first integral is easy
For the second, we need to use integration by parts:
Therefore, we have
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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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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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