Let f be the function defined by f(x) = −ln(x) for 0 < x ≤ 1. R is the region between the graph of f and the x-axis.

Question

Let f be the function defined by f(x) = −ln(x) for 0 < x ≤ 1. R is the region between the graph of f and the x-axis.



1. This question does not permit the use of a calculator. Determine whether region R has a finite area. If so, find the area. If not, explain why.
2. This question does not permit the use of a calculator. Determine whether the solid generated by revolving region R about the y-axis has a finite volume. If so, find the volume. If not, explain why.
3. The formula for the arc length from a to b of a differentiable function is . Find the length of the curve f(x) from x = 0.5 to x = 1. Your work must show the integral, but you may use your calculator to evaluate it. Give 3 decimal places in your answer.
4. g(x) = x². Find the area bounded by f(x), g(x) and the x-axis in the interval [0.5, 1]. Your work must show the integral, but you may use your calculator to evaluate it. Give 3 decimal places in your final answer.
5. Find the volume of the solid whose base is the region bounded by f(x), g(x) and the x-axis on the interval [0, 1], and whose cross-sections perpendicular to the y-axis are squares. Your work must show the integral, but you may use your calculator to evaluate it. Give 3 decimal places for your answer.

Answer 1

R=1.000

integral of lnx = xlnx-x

when you try to evaluate the integral at 0 to 1 you get an indeterminate form.

change xlnx to a fraction: lnx/(1/x) and apply L'Hopital's rule

then (0)ln(0) will equal 0

You might (wrongly) guess the area is infinite, but it's finite

My guess on the volume though is infinite, sort of like the volume of a cone would be infinite if it has infinite height and finite base area.