Florida B.E.S.T. MA.912.C.5.7

MathGrades 9–12Apply integrals to solve problems.

The Standard

Find the volume of a figure with known cross-sectional area, including figures of revolution, by using definite integrals.

Florida B.E.S.T. Standards for Mathematics

Teacher's field guide

What Students Need to Do: Students represent a solid as thin slices and write a formula for each slice's area. They choose the variable and bounds, then integrate to calculate volume.
What Mastery Looks Like: Students sketch the solid, identify a typical slice, and write its area in one variable. They use correct bounds and evaluate the integral with cubic units.
Common Misconceptions: Students may integrate a radius instead of a cross-sectional area. They often forget to square radii, include π, subtract the inner area, or use matching bounds and variables.
How to Assess It: Exit ticket: The region bounded by y = √x, y = 0, x = 0, and x = 4 rotates about the x-axis. Set up and evaluate its volume.

Lesson moves

Stack circular cardstock slices of changing radii, estimate the stack's volume, then connect the sum of slice volumes to a definite integral.
Ask students to explain when a rotated region creates disks rather than washers, using labeled sketches and radius expressions.
Run a card sort matching region graphs, rotation axes, cross-sectional shapes, bounds, and volume integrals.
Model a tapered paper cup with a linear radius function, then calculate its capacity and compare it with the printed volume.

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