Find area between $y=\frac1x,y=x,x=e$

Question

I have to calculate area between the 3 curves: $$y=\frac1x\quad y=x\quad x=e$$ I integrated $(x-1/x$) from 1 to $e$, but it does not match any option. The given answer was $\frac32$.

Can you explain the the approach?

Answer 1

If $y=1/x$ bounds the region from above and the $x$-axis is implied to bound the region from below, the given answer is obtained: $$\int_0^1x\,dx+\int_1^e\frac1x\,dx=\left[\frac{x^2}2\right]_0^1+[\ln x]_1^e=\left(\frac12-0\right)+(1-0)=\frac32$$ If $y=1/x$ bounds the region from below, leaving no implicit boundaries, a different answer is obtained: $$\int_1^e\left(x-\frac1x\right)\,dx=\left[\frac{x^2}2-\ln x\right]_1^e=\left(\frac{e^2}2-1\right)-\left(\frac12-0\right)=\frac{e^2-3}2$$ Good questions never leave anything to the reader to interpret, though. As egreg pointed out, the second answer has fewer assumptions and is the correct one.

Answer 2

you have 2 parts

from 0 to 1

&

from 1 to e

try to calculate the area between x and the x-axis

then between 1/x and x-axis