Find the area between the curve y=In(x), the y-axis and the lines y=2 and y=4

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You need to evaluate the area of the region bounded by the given curves, hence, you need to evaluate the following definite integral, such that:

`int_2^4 f(y) dy`

`f(x) = y = ln x =>f(y) = x = e^y`

Substituting `e^y ` for `f(y)` yields:

`int_2^4 e^y dy = e^y|_2^4`

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You need to evaluate the area of the region bounded by the given curves, hence, you need to evaluate the following definite integral, such that:

`int_2^4 f(y) dy`

`f(x) = y = ln x =>f(y) = x = e^y`

Substituting `e^y ` for `f(y)` yields:

`int_2^4 e^y dy = e^y|_2^4`

Using the fundamental theorem of calculus yields:

`int_2^4 e^y dy = e^4 - e^2`

Converting the difference of squares into a product yields:

`A = (e^2 - e)(e^2 + e)`

Factoring out e yields:

`A = e^2(e - 1)(e + 1)`

Hence, evaluating the area of the region bounded by the given curves, yields `A = e^2(e - 1)(e + 1).`

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