Question

If $A(n)$ represents the area bounded by the curve $y=n \log _e x(n \in N)$ and $n>1$, $x$-axis and the lines $x =1$ and $x = e$. Then $A ( n )- A ( n -1)$ is equal to

• A. 1
• B. $n^2$
• C. ne
• D. $e$

Answer 1

Answer: (A)

$ A(n)=\int\limits_1^e x \log _e x d x, f(x)=n \log _e x $
$A(n)=n\left[x \log _e x-x\right]_1^c $
$=n[e \log e-e-(1 \log 1-1)]=n$
$\text { So, } A(n)-A(n-1) $
$=n-(n-1)=1$