# Area & Volume

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

**2) Area between curves with respect to dx: **

**Explanation:**Khan Video: Area between curves

Khan Video: Area between curves with multiple boundaries

**Practice Problems:**

**4) Area between curves: Mixed Review **

**Explanation:**Hollis: Area Between Curves

**Practice Problems:**AP Questions:

**5) Volume by cross section:**

**Explanation:**Perpendicular to the x-axis:

Websites:

Video: Hollis, University of Houston (slides 1-5)

**Practice Problems:**

**6) Volume: Revolved about the x-axis:**

**Explanation:**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"

**Practice Problems:**

**7) Volume: Revolved about the y-axis:**

**Explanation**Khan: Solid of Revolution (part 6) - revolution around the y-axis (x = 0)

**Practice Problems**

**9) Mixed Volume: Review**

**Explanation:**Notes | Annotated | website - text based explanation | website - some animation

Interactive website: revolution (Shodor - Java)

**Practice Problems:**Desmos Activity: Volumes by Revolution (teacher use)

**10) Review**

**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 c2010 (Form B) Q1 - area, revolve about

*y*= 8, and cross section2010 Q4 - area, revolve about

*y*= 7, and cross section are rectangles perpendicular to the*y*-axis2009 (Form B) Q4 - area, cross section, revolve about

*y*= 22009 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*-axis2008 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*-axis2005 (Form B) Q1 - area, revolve about the

*x*-axis, and cross-section2005 Q1 - 2 area questions and revolve about

*y*= -12004 (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*-axis2003 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 section2002 Q1 - area, revolve about

*y*= 4, and minimum (no graphs are provided)2001 Q1 - 2 area questions and one revolve about the

*x*-axis2000 Q1 - area, revolve about the

*x*-axis, and cross section1999 Q2 - area, revolve about the

*x*-axis, and equal volume when divided and revolved1998 Q1 - area, 2 equal areas, revolve about the

*x*-axis, and 2 equal volumes by revolutions (no graphs are provided)

**11) ****Cumulative Review **