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Calculus practice

Approximating Area under a Curve worksheets

Draw and calculate left-endpoint, right-endpoint and midpoint rectangle estimates for explicit polynomial, trigonometric, exponential and natural-log functions.

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Draw the stated rectangles, evaluate the function at the correct sample points and multiply the height sum by the common width.

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Connect the function to every rectangle

Each graph shows the actual function, interval and partitions. Students draw rectangles at the stated left endpoints, right endpoints or midpoints, then use the same sample values in the numerical estimate.

Learning objective

What this activity builds

Use a function graph and equal-width rectangles to approximate geometric area under a positive curve.

Worked example

Where is the sample point for a midpoint rectangle?

Halfway between the subinterval endpoints.

Evaluate the function at that midpoint; the rectangle runs from the subinterval's left boundary to its right boundary.

Teaching notes

Make the practice count

  • Mark every partition before drawing rectangles.
  • Match each rectangle height to its stated endpoint or midpoint.
  • Keep full calculator precision and round only the final estimate to three decimals.

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Choose 4 or 8 tasks, one or more sampling methods, 3 through 6 rectangles, and any function groups. With 4 tasks and all 5 groups, four balanced groups appear; change the seed for another selection.

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Math SheetNameScoreTeacherDateApproximating Area under a CurveDraw the stated rectangles, then use unrounded function values to calculate the area estimate. Give the final estimate to three decimal places.Use radians for sine and cosine. Keep full calculator values in the work and round only the final estimate to 0.001.1. Left endpoint; n = 5f(x) = 5 + ((x - 3)/2)^3Interval [1, 5]19/513/517/521/5507xf(x)Draw the rectangles on the graph.Width:Height sum and area estimate:2. Right endpoint; n = 4f(x) = 4 - ((x - 0)/2)^2Interval [-2, 2]-2-101204.5xf(x)Draw the rectangles on the graph.Width:Height sum and area estimate:3. Right endpoint; n = 4f(x) = 4 + ((x - 3)/1)^2Interval [2, 4]25/237/2406xf(x)Draw the rectangles on the graph.Width:Height sum and area estimate:4. Left endpoint; n = 5f(x) = 3 + 2*((x - 3)/2)^3Interval [1, 5]19/513/517/521/5506xf(x)Draw the rectangles on the graph.Width:Height sum and area estimate:math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateApproximating Area under a CurveDraw the stated rectangles, then use unrounded function values to calculate the area estimate. Give the final estimate to three decimal places.Use radians for sine and cosine. Keep full calculator values in the work and round only the final estimate to 0.001.5. Right endpoint; n = 4f(x) = 4 + ((x + 1)/2)^2Interval [-3, 1]-3-2-10106xf(x)Draw the rectangles on the graph.Width:Height sum and area estimate:6. Left endpoint; n = 5f(x) = 3 - 2*((x + 1)/1)^3Interval [-2, 0]-2-8/5-6/5-4/5-2/5006xf(x)Draw the rectangles on the graph.Width:Height sum and area estimate:7. Left endpoint; n = 4f(x) = 5 - 2*((x - 1)/2)^2Interval [-1, 3]-1012306xf(x)Draw the rectangles on the graph.Width:Height sum and area estimate:8. Right endpoint; n = 5f(x) = 5 - ((x + 3)/2)^3Interval [-5, -1]-5-21/5-17/5-13/5-9/5-107xf(x)Draw the rectangles on the graph.Width:Height sum and area estimate:math-sheet.comPage 2 of 2

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