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Math 175- June 19, 2008
Exam 1 Name
This test consists of 100 points and 5 pages, none of which is intentionally left blank. Take a few seconds right now to be sure you have all the pages. The point value of each question is to the left of the question number. Show all your work in the space provided. If you run out of room for an answer, continue working on the back of the page. Your answers must be justified by your work.
(10) 1.Find the area of the region bounded by the graphs of y = x^2 and y = 2x + 3
(10) 2.The base of a solid is the upper semi-circular disk bounded by the graph of y =
4 − x^2 and the x-axis. Cross-sections perpendicular to the base and to the x-axis are equilateral triangles. Set up and evaluate an integral whose value is the volume of the solid.
(10) 3.The region bounded by the graphs of y = x(x^2 − 4) and the x-axis is revolved about the x axis. Set up, but do not evaluate, an integral whose value is the volume of the resulting solid.
(15) 4.Set up and evaluate an integral whose value is the length of the curve
x = et^ sin(t), y = e^2 cos(t), 0 ≤ t ≤ π
(10) 7.A spring has a natural length of 20cm. If a 20-N force is required to keep it stretched to a length of 30cm, how much work is required to stretch it from 20cm to 25cm.
(15) 8.Find the moment about the x-axis, the moment about the y-axis and the center of mass of a thin plate of constant density δ covering the region in the first and fourth quadrants enclosed by the curves y = 1/(1 + x^2 and y = − 1 /(1 + x^2 ) and by the lines x = 0 and x = 1
(10) 9.Use integration by parts to evaluate
∫ (^1)
0
x^3 √ 4 + x^2
dx
