Find the exact area enclosed between the curve `y=e^(2x)` and the lines `y=1`  and ` x=2`.    

Expert Answers

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You need to find area between green (`y=1` ), red (`x=2` ) and blue (`y=e^(2x)` ). Hence you need to find limits of integration. The upper limit is red line that is `x=2`  and lower limit is the intercept of `y=1` and `y=e^(2x)`, hence `1=e^(2x) => x=0`. Now that we have our limits we only need to calculate the integral:

`int_0^2 (e^(2x)-1)dx=`

-1 is here because when we calculate the area under `y=e^(2x)` we need to subtract the area under `y=1`.

`(e^(2x)/2-x)|_0^2=e^4/2-2-e^0/2-0=e^4/2-5/2`

So your solution is `e^4/2-5/2`

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