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

Interesting question! hold on .... :)

Answer 2

The are of the region is given by \[A(m)=\int_m^{3m}\frac{1}{x}\;dx=\int_0^{3m}(1/x)dx-\int_0^m (1/x)dx\]
By the fundamental theorem of calculus \[A'(m)=\frac{1}{3m}\cdot 3-\frac{1}{m}=0\]
So the area doesn't change with respect to m.
a) is the answer.

Answer 3

thanks!:)