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 L equals the integral from a to b of the square root of the quantity 1 plus the square of f prime of x dx. 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.