Math 107 Fall ’12  Exam 1  Score:
Name:  TA’s Name:

Instructions: You must show supporting work to receive full and partial credits. No text book, notes, formula sheets allowed.

1(13pts) Use the definition to determine if the improper integral converge: ∫_{1}^{2} \frac{1}{x − 1}dx. (Show all

work.)

2(12pts) Some values of a function y = f(x) are given in the table below.

\begin{matrix}x & 1 & 1.5 & 2 & 2.5 & 3 \\ f(x) & 1 & 2.25 & 4 & 6.25 & 9 \end{matrix}

(a) Use the right point sum with two partition, R_{2}, to approximate ∫_{1}^{3} f(x)dx.

(b) Use the midpoint sum, M_{2}, to approximate ∫_{1}^{3} f(x)dx.

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3(15pts) Let R be the region in the first quadrant that is bounded by two curves: y = x^{3} and y = x. Sketch the region and set up a definite integral for its area. Do not evaluate the integral.

4(10pts) The base of a solid is a quarter disk bounded by x^{2} + y^{2} = 1, x ≥ 0, y ≥ 0. Each cross section of the solid that is perpendicular to the x-axis is a square as shown. Set up an integral for the volume of the solid. Do not evaluate the integral.

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5(10pts) Let r = f(θ) = θ be the Archimedes spiral. Sketch the curve for 0 ≤ θ ≤ π/2 and set up an integral for the area of the region that is bounded by the spiral between θ = 0 and θ = π/2. Do not evaluate the integral.

6(15pts) A doughnut solid is obtained by rotating the unit disk bounded by x^{2} + y^{2} = 1 around the vertical line x = 2. Set up a definite integral for the volume of the doughnut. Do not evaluate the integral.

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7(15pts) Set up an integral for the length of the curve y = x^{2} in the interval 0 ≤ x ≤ 1. Do not evaluate the integral.

8(10pts) Use the Comparison Test to determine whether the improper integral converge or diverge: (Hint: Use x ≥ \sqrt{x} for x ≥ 1. Show all work.)

∫_{2}^{∞} \frac{x − \sqrt{x}}{x^{2} − 1} dx

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2 point Bonus Question: The first day of this year’s Fall is . (Page 4. ... The End)