600-232 CALCULUS II SAMPLE GATEWAY TEST #1

1.

(10 points each) Let $f(x) = x^2 - 1/\sqrt{x}$, $x \in [1,3]$, and let S denote the solid of revolution generated when the area under the graph of f is revolved around the x-axis. Write down and simplify, but do not evaluate, integrals that give the following:

(a)

The area under the graph of f

(b)

The arclength of the graph of f

(c)

The volume of S.

(d)

The surface area of S.

2.

(20 points) Sketch the region bounded by the curves y = x² + 6x and y = 3x - 2, and find its area.

3.

(20 points) A solid sits on the (x,y)-plane, with its base bounded by the curves y = -x², y = 3x², x = -2 and x = 2. The cross-sections perpendicular to the x-axis are rectangles, whose height is half their base. What is the volume of the solid?

4.

(10 points) Suppose that a force moves an object along the x-axis from x=1 to x=e². If the force at any point on the axis is given by $F(x) = \ln x$, find the work done by the force.

5.

(10 points) Find the average value of the function f(x) = 2x² + 3 on the interval [0, 1].