Calculus
Area Between Curves worksheets
Practice exact area between polynomial curves with intersection-derived or supplied limits, x or y integration, and every order-changing crossing split correctly.
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Volume Using Cross Sections
Practice exact volumes whose square, rectangular, or right-triangular cross sections stand on widths through semicircular, semi-oval, and elliptical bases.
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Sketch each base and cross section, derive its area function, and find the exact volume.
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Customize this worksheetTurn base widths into cross-sectional areas
Each problem starts with a measured ellipse or upper half-ellipse. Students identify the exact slice width, apply the square, rectangle, or right-triangle area rule, and integrate the resulting quadratic.
Learning objective
What this activity builds
Derive a slice width and cross-sectional area function, then integrate it exactly over the base.
Worked example
If a square cross section has side w(x), what is its area?
A(x) = w(x)^2.
The base region supplies the segment length w(x); the named cross-sectional shape turns that length into an area.
Teaching notes
Make the practice count
- Mark whether slices are perpendicular to the x-axis or y-axis before finding bounds.
- Square the radical width before expanding the area function.
- For a right triangle, w and rw are perpendicular legs, so A = rw^2/2.
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Choose 2 or 4 problems. The height ratio applies only to rectangles and right triangles; squares always have side ratio 1.
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