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Let f and g be the functions given by #f(x)=e^x# and #g(x)=1/x#. What is the area of the region enclosed by the graphs of f and g between x=1 and x=2?

Answer 1

Luke Alvarez

#A=e^2-e-ln2# units

If we take a peek at the graphs of these functions, we'll see this:

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Answer 2

Luke Alvarez

To find the area of the region enclosed by the graphs of ( f(x) = e^x ) and ( g(x) = \frac{1}{x} ) between ( x = 1 ) and ( x = 2 ), you need to calculate the definite integral of the difference of the two functions over the given interval. The integral represents the net area between the two curves.

[ \text{Area} = \int_{1}^{2} (f(x) - g(x)) , dx ]

[ = \int_{1}^{2} (e^x - \frac{1}{x}) , dx ]

[ = \left[ e^x - \ln|x| \right]_{1}^{2} ]

[ = (e^2 - \ln|2|) - (e^1 - \ln|1|) ]

[ = (e^2 - \ln(2)) - (e - \ln(1)) ]

[ = e^2 - e + \ln(1) - \ln(2) ]

[ = e^2 - e - \ln(2) ]

Therefore, the area of the region enclosed by the graphs of ( f ) and ( g ) between ( x = 1 ) and ( x = 2 ) is ( e^2 - e - \ln(2) ).

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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.