Q1 [25 pts]: Justify your answers clearly.

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

(−4)(+5)

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

Q2 [25 pts]: Justify your answers clearly.

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

−∞ or show that it is divergent.

- b) [8 pts] Evaluate the integral ∫

1−

1

0 or show that it is divergent.

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

of Type I.

Q3 [25 pts]: Justify your answers clearly.

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

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

Q4 [25 pts]: Justify your answers clearly.

- a) [10 pts] Find the volume of the solid generated by revolving the region bounded by = √, = 2, and = 0 about the -axis.

- b) [8 pts] Find the volume of the solid generated by revolving about the -axis the region bounded by = 4 − 2 and the -axis.
- c) [7 pts] A region between two curves (where ≥ 0) is rotated about the -axis. The cross-sections of the resulting object are as follows. Draw the region that is rotated and write the integral which gives the volume.