Answer to Question #27113 in Integral Calculus for jan

Question

find the area bounded by y=Inx,y=0 and x=e

Answer 1

The graph of the function y=ln(x) intersects x-axis atthe point x=1.

Therefore the area between these curves is equal to

A = int_{1}^{e} (ln(x) - 0) dx =
= int_{1}^{e}ln(x) dx

Let us use integrating by parts.
Notice that

ln(x) dx = d( x ln(x) ) - x d(ln(x)) =
= d( xln(x) ) - x * 1/x
= d( xln(x) ) - 1

Hence
A = int_{1}^{e} ln(x) dx
= x ln(x)|_{1}^{e} - int_{1}^{e} 1 dx
= e*ln(e) -1*ln(1) - (e-1) =
= e - 0 - e + 1
= 1