1. Section 7.2 Day 5 Cross Sections
AP Calculus AB

2. Learning Targets
Determine the volume of a region with square cross sections
Determine the volume of a region with rectangular cross sections
Determine the volume of a region with triangular cross sections (equilateral/other)
Determine the volume of a region with semi- circular cross sections

3. Square Cross Sections - Conceptual
Area of a square: 𝐴= 𝑠 2
Side Length: Area of the region - 𝑓(𝑥)
Volume of one cross section: 𝑓 𝑥 2 ∆𝑥
Volume of square cross sections:
𝑉= 𝑎 𝑏 𝑓 𝑥 2 𝑑𝑥
Perpendicular to x-axis: in terms of x
Perpendicular to y-axis: in terms of y

4. Rectangular Cross Sections - Conceptual
Area of a Rectangle: 𝐴=𝑏ℎ
Base Length: Area of the region - 𝑓(𝑥)
Volume of one cross section: 𝑓 𝑥 ℎ∆𝑥
For a rectangle, height will be specified
Volume of rectangular cross sections:
𝑉= 𝑎 𝑏 𝑓 𝑥 ∙ℎ 𝑑𝑥

5. Triangular Cross Sections - Conceptual
Area of an equilateral triangle: 𝐴= 𝑠 2
Side Length: Area of the region - 𝑓(𝑥)
Volume of one cross section: 𝑓 𝑥 2 ∆𝑥
Volume of triangular cross sections:
𝑉= 𝑎 𝑏 𝑓 𝑥 2 𝑑𝑥

6. Semi-Circle Cross Sections - Conceptual
Area of a semi-circle: 𝐴= 1 2 𝜋 𝑟 2
Radius Length: Area of the region - 𝑓(𝑥) (sometimes half of the area of the region)
Volume of one cross section: 1 2 𝜋𝑓 𝑥 2 ∆𝑥 or 𝜋 𝑓 𝑥 ∆𝑥
Volume of semi-circle cross sections:
𝑉= 𝑎 𝑏 1 2 𝜋𝑓 𝑥 2 𝑑𝑥 or 𝑉= 𝑎 𝑏 1 2 𝜋 𝑓 𝑥 𝑑𝑥

7. Exit Ticket
Describe the difference between disk, washer, and cross section methods of finding the volume of an object