Section 7.2 Day 5 Cross Sections AP Calculus AB

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

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

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

Triangular Cross Sections - Conceptual Area of an equilateral triangle: 𝐴= 3 4 𝑠 2 Side Length: Area of the region - 𝑓(𝑥) Volume of one cross section: 3 4 𝑓 𝑥 2 ∆𝑥 Volume of triangular cross sections: 𝑉= 𝑎 𝑏 3 4 𝑓 𝑥 2 𝑑𝑥

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 1 2 𝜋 𝑓 𝑥 2 2 ∆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 2 2 𝑑𝑥

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