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Grade 12 · Math · Calculus

12th grade Approximate area under a curve worksheet

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Approximate area under a curve

Grade 12 Math Worksheet - Challenge Me

NameDate

Circle the best answer, or write your answer on the line.

  1. Esme's function f rises all the way from x = 0 to x = 4. Is a left sum for the area too big or too small?

    • It is an underestimate: the left ends are the lowest points of each strip
    • It is an overestimate: the left ends are the highest points of each strip
    • It is exact: each rectangle touches the curve at a corner

  2. Jun's function f falls all the way from x = 1 to x = 5. Is a right sum for the area too big or too small?

    • It is an underestimate: the right ends are the lowest points of each strip
    • It is an overestimate: the right ends are the highest points of each strip
    • It is exact: each rectangle touches the curve at a corner

  3. Kira's function f rises all the way from x = 2 to x = 8. Is a right sum for the area too big or too small?

    • It is exact: each rectangle touches the curve at a corner
    • It is an overestimate: the right ends are the highest points of each strip
    • It is an underestimate: the right ends are the lowest points of each strip

  4. Read this first

    Speed of the bike (example data)

    time (s)speed (m/s)
    02
    54
    155
    307

  5. Milo records the speed of a toy bike at uneven times (see the table). Estimate the distance in meters with a left sum. What is it?

  6. Speed of the kite (example data)

    time (s)speed (m/s)
    01
    23
    64
    126

  7. Ren records the speed of a toy kite at uneven times (see the table). Estimate the distance in meters with a left sum. What is it?

  8. Speed of the car (example data)

    time (s)speed (m/s)
    05
    208
    3010
    6012

  9. Zee records the speed of a toy car at uneven times (see the table). Estimate the distance in meters with a left sum. What is it?

  10. Tala's rising function has f(1) = 1 and f(5) = 25. With 8 equal strips on that interval, how much bigger is the right sum than the left sum?

  11. Pia's rising function has f(0) = 2 and f(6) = 20. With 3 equal strips on that interval, how much bigger is the right sum than the left sum?

  12. Bo's rising function has f(2) = 5 and f(4) = 13. With 5 equal strips on that interval, how much bigger is the right sum than the left sum?

  13. Ari's rising function has f(−2) = 1 and f(4) = 10. With 3 equal strips on that interval, how much bigger is the right sum than the left sum?

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12th grade Approximate area under a curve worksheet – printable
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About this worksheet

Estimate the area under a curve with rectangles and trapezoids. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can find the width of each strip: (b − a) ÷ n.
  • I can find left, right and midpoint sums from a table or a formula.
  • I can find a trapezoid estimate, counting inside heights twice.
  • I can say whether a sum is too big or too small for a rising or falling curve.

Standard: none (beyond CCSS)

Words to know

strip: One thin piece of the region, width (b − a) ÷ n.

left (right) sum: Rectangles whose heights come from each strip's left (right) end.

midpoint sum: Rectangles whose heights come from each strip's middle.

trapezoid rule: Strips topped by slanted lines: width × (left + right) ÷ 2.

Common mistakes

Took each height at the right end of its strip. A left sum uses the left end of each strip.

Took each height at the left end of its strip, but this sum uses a different point.

Took each height at the middle of its strip, which is the midpoint sum.

Questions parents ask

What is the most common mistake with approximate area under a curve?

Took each height at the right end of its strip. A left sum uses the left end of each strip.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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