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Calculus practice

Area Under a Curve

Practice exact total geometric area for polynomial, trigonometric, and exponential functions across supplied intervals.

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Find total geometric area between the function and the x-axis. Split at every interior zero and add positive segment areas.

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Split before integrating

Every interior zero separates sign-consistent regions. Students evaluate each segment and add positive geometric areas.

Learning objective

What this activity builds

Find every interior zero, integrate each sign-consistent segment, and add positive areas.

Worked example

Why check each interval separated by a zero?

Check the sign on each side; split wherever needed so below-axis areas do not cancel above-axis areas.

A signed integral allows below-axis area to cancel above-axis area; total geometric area adds the positive magnitude from each sign-consistent segment.

Teaching notes

Make the practice count

  • Mark every zero inside the supplied interval.
  • Use −f(x) where the graph lies below the axis.
  • Keep trigonometric and exponential answers exact.

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Math SheetNameScoreTeacherDateArea Under a CurveFind total geometric area between the function and the x-axis. Split at every interior zero and add positive segment areas.1. Find total geometric area on [-1, 2].f(x) =x2-x-34-12xyShade, split at zeros, and add positive areas:2. Find total geometric area on [-5/2, 1/2].f(x) =12·x3+32·x2+x-5/21/2xyShade, split at zeros, and add positive areas:3. Find total geometric area on [3/2, 5/2].f(x) =-12·x2+123/25/2xyShade, split at zeros, and add positive areas:4. Find total geometric area on [7/2, 4].f(x) =-12·x3+3·x2-112·x+37/24xyShade, split at zeros, and add positive areas:Exact total area; square units.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateArea Under a CurveFind total geometric area between the function and the x-axis. Split at every interior zero and add positive segment areas.5. Find total geometric area on [-2, -1].f(x) =12·x2+32·x+98-2-1xyShade, split at zeros, and add positive areas:6. Find total geometric area on [-7/2, -3/2].f(x) =x3+7·x2+16·x+12-7/2-3/2xyShade, split at zeros, and add positive areas:7. Find total geometric area on [-1/2, 5/2].f(x) =-x2+2·x-1/25/2xyShade, split at zeros, and add positive areas:8. Find total geometric area on [-5/2, 1/2].f(x) =-x3-3·x2-2·x-5/21/2xyShade, split at zeros, and add positive areas:Exact total area; square units.math-sheet.comPage 2 of 2

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