Calculus 2 problem
Subjective Short Answer
1. Find if .
2. Find an equation of the tangent line to the graph of at the point .
3. Evaluate the using L’Hôpital’s Rule if necessary.
4. Use the differential equation and the initial condition to find y.
Objective Short Answer
5. Evaluate the definite integral.
6. Find the following indefinite integral.
7. Evaluate the limit using L'Hopital's Rule if necessary.
8. Find an equation of the tangent line to the graph of at the point .
9. Evaluate the integral .
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10. Find the area of the region bounded by the graphs of the algebraic functions.
11. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x- axis.
, , ,
12. Use the shell method to set up and evaluate an integral that gives the volume of the solid generated by revolving the plane region about the y-axis.
,
13. Use the shell method to find the volume of the solid generated by revolving the plane region bounded by ,
, about the line .
14. Find the arc length of the graph of the function over the interval .
15. Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of ,
about the x-axis.
16. Find the indefinite integral.
17. Find the indefinite integral.
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18. Evaluate the definite integral .
19. Find the indefinite integral.
20. Find the indefinite integral.
21. Use partial fractions to find .
22. Use partial fractions to find the integral .
23. Use integration tables to evaluate the integral .
24. Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.
25. Find the derivative of the function .
26. Find the indefinite integral.
27. Solve the following equation for .
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28. Find if .
29. Evaluate the limit first by using techniques from Chapter 1 then by using L'Hopital's Rule.
30. Evaluate the limit using L'Hopital's Rule if necessary.
31. Evaluate the limit using L'Hopital's Rule if necessary.
32. Evaluate the expression without using a calculator.
33. Find the derivative of the function .
34. Find the integral .
35. Find the indefinite integral.
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36. Find the area of the shaded region for the function .
37. Find the area of the region bounded by the equations by integrating (i) with respect to x and (ii) with respect to y.
38. Find the area of the region bounded by the graphs of the equations.
, ,
39. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by
and in the first quadrant about the y-axis.
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40. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line .
41. Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.
, ,
42. Use the disk or shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line .
, , ,
43. Find the arc length of the graph of the function over the interval .
44. Find the area of the surface generated by revolving the curve about the x-axis.
, .
45. Find the arc length of the graph of the function over the interval .
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46. Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of about
the y-axis. Round your answer to three decimal places.
47. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.
(i) x-axis; (ii) the line
48. Find the integral
49. Find the slope-intercept form of the equation of the line tangent to the graph of when .
50. Write the following expression in algebraic form.
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Answer Key
1.
2.
3. 0
4.
5.
6.
7. 1
8.
9.
10.
11.
12.
13.
14.
15.
16.
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17.
18. 0
19.
20.
21.
22.
23.
24. diverges
25.
26.
27.
28.
29. 1
30.
31. 0
32.
33.
34.
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35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46. 36.177
47.
48.
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49.
50.
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