TEST 2



MAT 271

Final Exam Review

Fall/2006

Dr. Firoz

1. Evaluate the integrals.

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

g) [pic] h) [pic] i) [pic]

j) [pic] k) [pic]

2. Prove the reduction formula[pic]. Hence evaluate[pic]

3. Use method of substitution and the following trigonometric result

[pic] to evaluate the integral[pic].

4. Use [pic]to simplify the integral [pic] and then use the formula [pic] for the integral result.

5. Integrate [pic] using partial fraction

6. Find the area of the region bounded by [pic] and [pic], and [pic]. Sketch the enclosed region. Draw a typical approximating rectangle and label its height and width.

7. The region is bounded by the curve [pic], y = 8, and x = 0. Express in definite integral the volume of the solid obtained by

a) Rotating the region about [pic]. (You need not evaluate the integral). Draw the region of the solid.

b) Rotating the region about y = - 2. (You need not evaluate the integral). Draw the region of the solid.

8. Use the method of cylindrical shell to find the volume of the solid bounded by the curves [pic] and [pic] rotated about x axis.

9. A spring has natural length of 20 cm. If 25-N force is required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 25 cm to 28 cm?

10. Draw the graph of [pic], consider the average value of f over the following intervals:

I) [pic] II) [pic] III) [pic] IV) [pic]

a) For which interval is the average value of f least?

b) For which interval is the average value of f greatest?

c) For which pair of intervals the average values equal?

11. Set up the integral for the length of the curve [pic]. Do not integrate.

12. Find the area of the surface obtained by rotating the curve [pic] about the x axis.

13. a) Represent the given Cartesian point [pic]in terms of polar coordinates. Also plot the point.

b) Consider the parametric curve [pic]. Determine [pic].

14. a) Find an equation of a tangent line to the curve [pic].

b) Find the length of the parametric curve [pic].The length is given by the formula [pic].

15. Write down the Cartesian form of the polar equation [pic].

16. Consider the parametric curve [pic]. Find the points on the curve where the tangent line is horizontal or vertical. Graph the curve along with horizontal, vertical tangent(s) you have.

17. Show that the polar equation of the given Cartesian equation [pic] is [pic].

18. Find the area of the region that lies inside the circle [pic] and outside the cardioid [pic]. Draw rough graph of the given curves and identify the region. [6 points]

19. Find the center, vertices, and foci of the ellipse [pic] . Sketch the ellipse.

20. Determine whether the sequence given by [pic] converges or diverges. If it converges, find the limit.

21. Determine whether the series [pic] is convergent or divergent. If it is convergent, find its sum.

22. Determine whether the series converges or diverges.

a) [pic] b) [pic]

c) [pic] d) [pic]

23. Find the radius of convergence and interval of convergence of the series

[pic]

24. a) Evaluate the indefinite integral [pic]as a power series:

b) Show that [pic], C is a constant.

25.

25. Find the Taylor series for [pic] centered at a = 3.

26. a) Express the integral [pic] as an infinite series.

b) How many terms are needed to approximate

Useful formulas:

[pic] [pic] [pic]

[pic] for x axis rotation [pic] for y axis rotation

[pic]

[pic]

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