Area & Volume

Curriculum Modules: Volumes of Solids by Revolution by the College Board (pdf - 16 pages)

1) Total Area

Area between curves:

2) With respect to dx:

3) With respect to dy:

4) Mixed Review

5) Volume by cross section:

Volume by revolution:

6) Revolved about the x-axis: Notes | Annotated | Worksheet | Answer

7) Revolved about the y-axis: Notes | Annotated | Worksheet | Answer

8) Revolved about any horizontal or vertical lines: Notes | Annotated

9) Mixed Review

10) Review

11) Cumulative Review WS | Answers

AP Questions

    • 2014 Q2 - Volume by revolution about y = -2; cross sections with right isosceles triangles; equal areas
  • 2012 Q2 - area, cross section of different upper limits, two equal regions
  • 2011 (Form B) Q3 - area, cross section (rectangle with height 2y)
  • 2011 Q3 - area and revolve about y = 1; Khan video: part a & b | part c
  • 2010 (Form B) Q1 - area, revolve about y = 8, and cross section
  • 2010 Q4 - area, revolve about y = 7, and cross section are rectangles perpendicular to the y-axis
  • 2009 (Form B) Q4 - area, cross section, revolve about y = 2
  • 2009 Q4 - area & two cross section problems (one is a function and the other is dy)
  • 2008 (Form B) Q1 - area, revolved about x = -1, cross-section perpendicular to the y-axis
  • 2008 Q1 - 2 area questions and 2 cross sections (one depth is a function)
  • 2007 (Form B) Q1 - 2 area questions and one revolve about y = 1
  • 2007 Q1 - area, revolution about x-axis, and cross section (no graphs are provided)
  • 2006 (Form B) Q1 - 2 area and revolve about y = - 2
  • 2006 Q1 - area, revolve about y = - 3 and the y-axis
  • 2005 (Form B) Q1 - area, revolve about the x-axis, and cross-section
  • 2005 Q1 - 2 area questions and revolve about y = -1
  • 2004 (Form B) Q1 - area, revolve about y = 3 and x = 10 (no graph provided)
  • 2004 Q2 - area, revolve about y = 2, and cross section (challenging cross section)
  • 2003 (Form B) Q1 - area and revolve about the x-axis
  • 2003 Q1 - area, revolve about y = 1, and cross section (height is 5 times its base)
  • 2002 (Form B) Q1 - area, revolve about the x-axis, and cross section
  • 2002 Q1 - area, revolve about y = 4, and minimum (no graphs are provided)
  • 2001 Q1 - 2 area questions and one revolve about the x-axis
  • 2000 Q1 - area, revolve about the x-axis, and cross section
  • 1999 Q2 - area, revolve about the x-axis, and equal volume when divided and revolved
  • 1998 Q1 - area, 2 equal areas, revolve about the x-axis, and 2 equal volumes by revolutions (no graphs are provided)