Area Under Curves

Question

Area bounded by the curves $y = \ln x, y = 0$, and $x = e$ is:

RankGuru · Accepted Answer

1. Setup Integral:
$y = \ln x$ intersects $y = 0$ at $x = 1$.
The region is from $x = 1$ to $x = e$.
Area $= \int_1^e \ln x dx$.

2. Evaluate:
Using integration by parts ($\int u dv$), let $u=\ln x, dv=dx$.
$\int \ln x dx = x \ln x - x$.

Apply limits:
$[x \ln x - x]_1^e$
$= (e \ln e - e) - (1 \ln 1 - 1)$
$= (e - e) - (0 - 1)$
$= 0 - (-1) = 1$.

Rank Guru