Let R be the region bounded by the curve y = ln (x), the x-axis and the line x = e. Find the area of R. I need a quick answer with minimal work shown. Thanks

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Let R be the region bounded by the curve y = ln (x), the x-axis and the line x = e. Find the area of R. I need a quick answer with minimal work shown. Thanks

Mathematics
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Is there a graph you are suppose to give us?
That's all the question says.

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ok, so you have to integrate that from x=1 to x=e
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what I drew, is the picture, and x=1 is the limit of integration, because that is where the area above the x-axis starts, and x=e is the limit of integration because your region is bounded by x=e.
the regeion is from x=1 to x=e
so do integral of lnx from 1 to e
\(\Large\color{slate}{\displaystyle\int\limits_{1}^{e}{\rm Ln}(x)~dx}\) like this.
So just set it up and solve it
yup
?
you know the integral of ln(x) right?
Yes xlnx - x
yup, that is right. Now you have to plug in the limits of integration.
\(\Large\color{slate}{\displaystyle\int\limits_{1}^{e}{\rm Ln}(x)~dx=\left(x{\rm Ln} {\tiny~}x~-~x\right)~{\Huge|}_{1}^{e}}\)
Yeah I got that now thanks. How would I find the volume of the same problem on the x axis and also on the y axis?
Volume? Do you mean that the region we drew is rotated around some line?
Yeah I got that now thanks. How would I find the volume of the same problem on the x axis and also on the y axis?
Yep exactly that
you mean the area, not the volume? if volume, then it is 3d, and probably this R is rotated around some line y=C or x=C. if area, then what area do you exactly what to find?

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