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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].


 

Kevin B Mcleod
2/18/2002