User:Melikamp/calc
Review of Anti-derivatives edit
1. Find
2. Find
3. Find
4. Find
5. Find
6. Find
Integration by Parts edit
10. Find
11. Find
12. Find
Integration by Partial Fractions edit
13. Find
14. Find
15. Find
Improper Integrals edit
20. Find
21. Find
22. Find
23. Find
24. Find
Integration Review edit
30. Find
31. Find
32. Find
33. Find
34. Find
Distance Traveled and Arc Length in Space edit
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
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
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
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.
71. Find the mass of the thin plate lying in the -plane below the curve and above the curve if the area density is .
Center of Mass and Moments edit
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
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
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
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
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
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 .
- Find the direction in which the directional derivative of at is maximized, and the value of .
- Find the directions for which the directional derivative of at is zero.