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Calculus practice

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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Find the total area between the curves. Split the interval wherever the curve order changes.

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Prevent signed cancellation

Each worked key partitions the limits at every interior intersection, identifies the curve order, and integrates a positive difference on that segment.

Learning objective

What this activity builds

Identify the upper and lower or right and left curve on each exact subinterval, integrate a positive difference, and add the segment areas.

Worked example

What happens when the curves cross inside the interval?

Split at the crossing and integrate the positive curve difference on each side.

One signed integral can cancel real area, so each order-consistent segment contributes a positive area.

Teaching notes

Make the practice count

  • Mark every intersection inside the limits before integrating.
  • For dy, use right minus left instead of upper minus lower.
  • Add exact positive segment areas before attaching square units.

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Math SheetNameScoreTeacherDateArea Between CurvesFind the total area between the curves. Split the interval wherever the curve order changes.1. Integrate with respect to x.y =2·x2-x-2y =x2-x-1Find the intersection limits and all enclosed regions.-101-2-11fgxyShade the region, split if needed, and integrate:2. Integrate with respect to x.y =-2·x-2y =12·x2-74·x-94Find the intersection limits and all enclosed regions.-1-0.500.5-3-2-1fgxyShade the region, split if needed, and integrate:3. Integrate with respect to x.y =-2·x+1y =12·x3+74·x2-34·x+1Find the intersection limits and all enclosed regions.-2-10246fgxyShade the region, split if needed, and integrate:4. Integrate with respect to x.y =x+2y =x2-52·x+92Find the intersection limits and all enclosed regions.11.522.52.533.544.5fgxyShade the region, split if needed, and integrate:Exact total area; square units.math-sheet.comPage 1 of 1

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