Question

A region in the plane is bounded by the graph of y=1/x, the x-axis, the line x=m and the line x=3m, m>0. the area of this region:
a) is independent of m
b) increases as m increases
c) decreases as m increases
d) decreases for all m<1/3
e) increases for all m<1/3

Answer 1

Answer:

The area of plane is independent of m, A=log 3

A is correct

Step-by-step explanation:

A region in the plane is bounded by the graph of

$y=\frac{1}{x}$

x-axis

Line: x=m and x=3m

Please find the attachment for figure.

Area under the curve = integrate the curve from m to 3m

$A=\int_{m}{3m}\frac{1}{x}dx$

$A=\log x|_{m}^{3m}$

$A=\log 3m-\log m$

Using log property: log a - log b = log(a/b)

$A=\log(\frac{3m}{m}$

$A=\log 3$

Hence, The area of plane is independent of m.

Answer 2

The correct option is a): "is independent of m".
 
By the fundamental theorem of calculus:A′(m)=13m⋅3−1m
         =0 So the area doesn't change with respect to m.