Question

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

Answer 1

Is there a graph you are suppose to give us?

Answer 2

That's all the question says.

Answer 3

ok, so you have to integrate that from x=1 to x=e

Answer 4

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.

Answer 5

the regeion is from x=1 to x=e

Answer 6

so do integral of lnx from 1 to e

Answer 7

\(\Large\color{slate}{\displaystyle\int\limits_{1}^{e}{\rm Ln}(x)~dx}\) like this.

Answer 8

So just set it up and solve it

Answer 9

yup

Answer 10

?

Answer 11

you know the integral of ln(x) right?

Answer 12

Yes xlnx - x

Answer 13

yup, that is right.
Now you have to plug in the limits of integration.

Answer 14

\(\Large\color{slate}{\displaystyle\int\limits_{1}^{e}{\rm Ln}(x)~dx=\left(x{\rm Ln} {\tiny~}x~-~x\right)~{\Huge|}_{1}^{e}}\)

Answer 15

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?

Answer 16

Volume? Do you mean that the region we drew is rotated around some line?

Answer 17

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?

Answer 18

Yep exactly that

Answer 19

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?