Question

A region in the $x y$ plane is bounded by the graph of $y=\frac{1}{x}$, the $x$-axis, the line $x=m$, and the line $x=2 m , m >0$. The area of this region

• A. is independent of $m$
• B. increases as $m$ increases
• C. decreases as $m$ increases
• D. decreases as $m$ increases when $m <\frac{1}{2}$ and increases as $m$ increases when $m >\frac{1}{2}$

Answer 1

Answer: (A)

Area $=\int\limits_{ m }^{2 m } \frac{1}{ x } dx =\left.\ln x \right|_{ m } ^{2 m }$
$=\ln (2 m )-\ln ( m )=\ln 2$, so the area is independent of m