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Areas and Volumes Practice Problems for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

Review the following concepts if needed:

Part A–The use of a calculator is not allowed.

  1. Let where the graph of f is given in Figure 12.7-1.
  2. Areas and Volumes Practice Problems

    1. Evaluate F (0), F (3), and F (5).
    2. On what interval(s) is F decreasing?
    3. At what value of t does F have a maximum value?
    4. On what interval is F concave up?
  3. Find the area of the region(s) enclosed by the curve f (x) = x3, the x-axis, and the lines x = – 1 and x =2.
  4. Find the area of the region(s) enclosed by the curve y = |2x – 6|, the x-axis, and the lines x = 0 and x = 4.
  5. Find the approximate area under the curve from x = 1 to x = 5, using four right-endpoint rectangles of equal lengths.
  6. Find the approximate area under the curve y = x2 + 1 from x = 0 to x = 3, using the Trapezoidal Rule with n = 3.
  7. Find the area of the region bounded by the graphs
  8. Find the area of the region bounded by the curves x = y2 and x = 4.
  9. Find the area of the region bounded by the graphs of all four equations:
  10. Find the volume of the solid obtained by revolving about the x-axis, the region bounded by the graphs of y = x2 + 4, the x-axis, the y-axis, and the lines x = 3.
  11. The area under the curve from x = 1 to x = k is 1. Find the value of k.
  12. Find the volume of the solid obtained by revolving about the y-axis the region bounded by x = y2 + 1, x = 0, y = – 1, and y = 1.
  13. Let R be the region enclosed by the graph y = 3x, the x-axis and the line x = 4. The line x = a divides region R into two regions such that when the regions are revolved about the x-axis, the resulting solids have equal volume. Find a.
  14. Part B–Calculators are allowed.

  15. Find the volume of the solid obtained by revolving about the x-axis the region bounded by the graphs of f (x) = x3 and g (x) = x2.
  16. The base of a solid is a region bounded by the circle x2 + y2 = 4. The cross of the solid sections are perpendicular to the x-axis and are equilateral triangles. Find the volume of the solid.
  17. Find the volume of the solid obtained by revolving about the y-axis, the region bounded by the curves x = y2 and y = x – 2.
  18. For Problems 16 through 19, find the volume of the solid obtained by revolving the region as described below. See Figure 12.7-2.

    Areas and Volumes Practice Problems

  19. R1 about the x-axis.
  20. R2 about the y-axis.
  21. R1 about the line .
  22. R2 about the line
  23. The function f (x) is continuous on [0, 12] and the selected values of f (x) are shown in the table.
  24. Find the approximate area under the curve of f from 0 to 12 using three midpoint rectangles.

  25. Find the area bounded by the curve defined by x = 2 cos t and y = 3 sin t from t = 0 to t = π.
  26. Find the length of the arc of r = sin2 from θ = 0 to θ = π.
  27. Find the area of the surface formed when the curve defined by x = et sin t and y =et cos t from t = 0 to is revolved about the x-axis.
  28. Find the area bounded by r = 2 + 2 sin θ.
  29. The acceleration vector for an object is . Find the position of the object at t = 1 if the initial velocity is and the initial position of the object is at the origin.
  30.  

    (Calculator) indicates that calculators are permitted.

  31. in terms of k.
  32. A man wishes to pull a log over a 9 foot high garden wall as shown. See Figure 12.8-1. He is pulling at a rate of 2 ft/sec. At what rate is the angle between the rope and the ground changing when there are 15 feet of rope between the top of the wall and the log?
  33. Areas and Volumes Practice Problems

  34. (Calculator) Find a point on the parabola that is closest to the point (4, 1).
  35. The velocity function of a particle moving along the x-axis is v(t) = t cos(t2 + 1) for t ≥ 0.
    1. If at t = 0, the particle is at the origin, find the position of the particle at t = 2.
    2. Is the particle moving to the right or left at t = 2?
    3. Find the acceleration of the particle at t =2 and determine if the velocity of the particle is increasing or decreasing. Explain why.
  36. (Calculator) given f (x) = xex and g (x) = cos x, find:
    1. the area of the region in the first quadrant bounded by the graphs f (x), g (x), and x = 0.
    2. The volume obtained by revolving the region in part (a) about the x-axis.
  37. Find the slope of the tangent line to the curve defined by r = 5 cos 2θ at the point where

Solutions for these practice problems can be found at: Solutions to Areas and Volumes Practice Problems for AP Calculus

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