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# Analytic Geometry and Calculus II

1. a) [8 pts] Evaluate the integral ∫ (−3)

(−4)(+5)

1. b) [10 pts] First use a substitution and then partial fractions to evaluate the integral ∫ 1

√+1

1. c) [7 pts] When we calculate the integral of a rational function by partial fractions, in which

case(s) we get an inverse tangent function in the end? Explain your answer.

1. a) [10 pts] Evaluate the integral ∫ −4 1

−∞ or show that it is divergent.

1. b) [8 pts] Evaluate the integral ∫

1−

1

0 or show that it is divergent.

1. c) [7 pts] what is an improper integral of Type I? Give an example and explain why it is improper

of Type I.

1. a) [10 pts] Sketch the region enclosed by the curves = , = 1/2, and the line = 2 and find its area.
2. b) [8 pts] Find the area between the curves = 2 and = 3.
3. c) [7 pts] Does the integral ∫ ( − 2) 1

0 represent the area of a region? If so, make a sketch of the region.