# 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: **

- Explanation:
- Notes | Annotated
- Khan Video: Area between curves
- Khan Video: Area between curves with multiple boundaries

- Practice Problems:

** 3) With respect to dy**:

** 4) Mixed Review **

- Videos:
- Hollis: Area Between Curves

- Practice Problems:

**5) Volume by cross section:**

- Explanation:
- Perpendicular to the x-axis:
- Websites:
- Video: Hollis, University of Houston (slides 1-5)
- Isometric Paper

- Practice Problems:

**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

- Notes | Annotated | website - text based explanation | website - some animation
- Desmos Activity: Volumes by Revolution (teacher use)
- Practice Problems:
- Interactive websites:
- Interactive website: revolution (Shodor - Java)

- Videos:
- Hollis, University of Houston
- Khan: Solid of Revolution (part 1) - revolved around the x-axis (y = 0)
- Khan: Solid of Revolution (part 2) - revolved around the x-axis (y = 0)
- Khan: Solid of Revolution (part 4) - more revolution around the x-axis (y=0) - "washers"
- Khan: Solid of Revolution (part 6) - revolution around the y-axis (x = 0)

**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)