User:Melikamp/calc

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Review of Anti-derivatives[edit | edit source]

1. Find
2. Find
3. Find
4. Find
5. Find
6. Find

Integration by Parts[edit | edit source]

10. Find
11. Find
12. Find

Integration by Partial Fractions[edit | edit source]

13. Find
14. Find
15. Find

Improper Integrals[edit | edit source]

20. Find
21. Find
22. Find
Diverges
Diverges
23. Find
24. Find

Integration Review[edit | edit source]

30. Find
31. Find
32. Find
33. Find
34. Find

Distance Traveled and Arc Length in Space[edit | edit source]

40. Find the distance traveled by the particle with position function for .
41. Find the distance traveled by the particle with position function for .
42. Find the distance traveled by the particle with position function for .
43. Find the arc length of the graph of the function , where , for .
44. Find the arc length of the graph of the function for

Area Swept Out[edit | edit source]

50. Find the area swept out by a particle moving along the parametrized curve , , for . Plot the curve and shade the area swept out before setting up the integral.
51. Find the area swept out by a particle moving along the parametrized curve , , for . Plot the curve and shade the area swept out before setting up the integral.
52. Plot the polar curve for , and find the area enclosed by it.
53. Plot the polar curve for , and find the area enclosed by it.
54. Plot the polar curve for , and find the area enclosed by it.

Volume[edit | edit source]

60. Let be the region in the first quadrant above the -axis and below the curve , , a positive ingeter. Find the volume of the solid obtained by revolving about the -axis.
61. Let be the region above the line and below the line , . Find the volume of the solid obtained by revolving about the -axis.

Mass and Density[edit | edit source]

70. Find the mass of a stick extending along the -axis from to if the linear density of the stick is (assume SI units). Write an equation for the mass midpoint.
Mass is kg, mass midpoint is at , where
Mass is kg, mass midpoint is at , where
71. Find the mass of the thin plate lying in the -plane below the curve and above the curve if the area density is .
kg
kg

Center of Mass and Moments[edit | edit source]

80. Find the center of mass of a thin wire extending from to along the x-axis if the linear density of the wire is .
81. Find the center of mass of a thin plate occupying the region in the xy-plane, if is a region below the curve and above the curve , with , and the area density of the plate is .

Work and Energy[edit | edit source]

90. Suppose that we have a tank, which is a right circular cylinder of radius 1 meter and height 4 meters, and the tank is initially filled half-way. Find the amount of work required to pump all of the water to the top of the tank. Use as the density of water and as the gravity of Earth.
91. Suppose that a bucket is lifted to the top of a building 12 meters high at a constant rate of . The initial weight of the bucket is , and it is leaking sand at the rate of . Find the work required to lift the bucket. Use as the gravity of Earth.

Taylor Series[edit | edit source]

100. Find the Taylor polynomial of 8th degree, centered at zero, for the function , and make a guess about the corresponding Taylor series.
101. Find the Taylor polynomial of 8th degree, centered at zero, for the function , and make a guess about the corresponding Taylor series.
102. Find the Taylor series for the function centered at .
103. Find the Taylor polynomial of 3rd degree for the function , centered at .

Cross Product[edit | edit source]

110. Find the area of the triangle with vertices , , and .
111. Find a standard equation for the plane containing both the point and the line .

Multi-Component Functions of a Single Variable[edit | edit source]

120. Let the position of the particle be given by . Find velocity, acceleration, and speed of the particle as functions of . Sketch the path of the particle, and draw the velocity vector and the acceleration vector .
, ,
, ,
121. Find the distance traveled by a particle between times and if the position function of the particle is .

Directional Derivative[edit | edit source]

130. Find the total derivative of at the point , if .
131. Find the value of the directional derivative of at in the direction of the vector .
132. Let and .
  1. Find the direction in which the directional derivative of at is maximized, and the value of .
  2. Find the directions for which the directional derivative of at is zero.
(a) and , (b)
(a) and , (b)