A region in the x y plane is bounded by the graph of y=(1/x), the x-axis, the line x=m, and the line x=2 m , m 0. The area of this region

Question

A region in the x y plane is bounded by the graph of y=(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 <(1/2) and increases as m increases when m >(1/2)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/0caaf317943cf6ca8b97fbcc195b8492-.png" /> Area =∫ limits m 2 m (1/ x ) dx =. ln x | m 2 m = ln (2 m )- ln ( m )= ln 2, so the area is independent of m

Q. 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

is independent of $m$

increases as $m$ increases

decreases as $m$ increases

decreases as $m$ increases when $m < \frac{1}{2}$ and increases as $m$ increases when $m > \frac{1}{2}$

Area $= m \int 2 m \frac{1}{x} d x = ln x ∣_{m}^{2 m}$
$= ln (2 m) - ln (m) = ln 2$ , so the area is independent of m

Q. A region in the xy plane is bounded by the graph of y=x1​, the x-axis, the line x=m, and the line x=2m,m>0. The area of this region

is independent of m

increases as m increases

decreases as m increases

decreases as m increases when m<21​ and increases as m increases when m>21​