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.

Last Updated on December 3, 2020 by Essay Pro