Lesson 7: 8.7 Cross Sections: Squares and Rectangles
Duration of Days: 2
Lesson Objective
The student will be able to:
Visualize a 3D solid rising out of a 2D bounded region on the xy-plane.
Define the "base" (s) of a cross section as the vertical or horizontal distance between two functions.
Construct an area function, A(x) or A(y), based on the geometry of the cross section.
Integrate the area function over a specific interval to calculate the total volume of the solid.
How can we transition from finding a "flat" area to finding a "thick" volume?
If the length of a square's side is determined by the distance between two curves, how does that side length change as we move along the x-axis?
Why is the volume of the entire solid equal to the integral of the area of a single representative slice?
Cross Section
Base of the Solid
Volume by Slicing
Integrand (Area Function)
Perpendicular to the x-axis (or y-axis)
Prism / Slice
CHA-5.B: Calculate volumes of solids with known cross sections.CHA-5.B.1: The volume of a solid with known cross-sectional area A(x) from x=a to x=b is the integral.
Mathematical Practice 1.E: Apply appropriate mathematical rules or procedures.
Mathematical Practice 4.C: Connect concepts to confirm or clarify the relationship between functions.
Description: This lesson introduces the "Volume by Slicing" method. Students start with a 2D region (the "base") and imagine shapes (squares or rectangles) "popping out" of the paper. They learn that the distance between the top and bottom functions, s = f(x) - g(x), becomes the side of the square or the base of the rectangle. By integrating the area of these shapes from start to finish, they find the volume.
Purpose: This is the first of three lessons on volume. It builds the conceptual framework that Volume = integral of an area slice. Mastering squares and rectangles first allows students to focus on the setup before moving to more complex geometry like semicircles and triangles.
DOK Level 2 (Skill/Concept): Identifying the side length s and plugging it into the volume formula.
DOK Level 3 (Strategic Thinking): Setting up volumes for rectangles where the height is given as a function of the base.
For Struggling Learners (Scaffolding):The "Bread Sieve" Analogy: Compare the solid to a loaf of bread. The integral is just adding up the volume of every individual slice.
Area-to-Volume Template: Provide a three-step table:
Find s (Top - Bottom).
Write A(x) (Area formula for the shape).
Integrate area.
Physical Manipulatives: Use 3D printed models or foam cutouts to show how a square sits on top of a 2D function.
For Advanced Learners (Extension):Perpendicular to the y-axis: Challenge students to set up the same volume but with slices perpendicular to the y-axis, requiring them to solve for x first.
Variable Heights: Provide problems where the height of the rectangle is not a multiple of the base, but a separate function entirely, h(x).
AP College Board Assessment