Worksheets · Calculus

Riemann sums from a table worksheet

Nothing about a Riemann sum needs a formula: a column of readings, evenly spaced or not, is enough. This is how the rules are actually used, because real measurements arrive as tables.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Riemann Sums From a Table

Date Period

Estimate each value from the table.

  1. Use a left sum from a table

    Oil leaks from a tank at a rate r in gallons per hour, measured at these times t in hours. Use a left Riemann sum to estimate how much oil leaks out from t=0 to t=8.
    t02468r1218242224
  2. Rain falls at a rate r in millimeters per hour, measured at these times t in hours. With a left Riemann sum, estimate how much rain falls between t=0 and t=20.
    t05101520r1621171520
  3. A runner's speed v in m/s is measured at these times t in seconds. Estimate how far he runs, using a left Riemann sum.
    t02468v46969
  4. A car's speed v in m/s is recorded at these times t in seconds. What estimate of how far it travels does a left Riemann sum give?
    t02468v711121311

  1. Use a right sum from a table

    Water flows into a tank at a rate r in liters per minute, recorded at these times t in minutes. What estimate of how much water flows in does a right Riemann sum give?
    t0510152025r722486
  2. A car's speed v in m/s is recorded at these times t in seconds. Use a right Riemann sum to estimate how far it travels from t=0 to t=10.
    t0246810v161925202624
  3. A cyclist's speed v in ft/s is recorded at these times t in seconds. Estimate how far she rides, using a right Riemann sum.
    t51015202530v77612129
  4. Water flows into a tank at a rate r in liters per minute, recorded at these times t in minutes. With a right Riemann sum, estimate how much water flows in between t=5 and t=30.
    t51015202530r121318161116

  1. Use the trapezoidal rule from a table

    A house draws electric power P in kilowatts, read at these times t in hours. Estimate how much energy it uses, using the trapezoidal rule.
    t12345P1218211720
  2. A file downloads at a rate r in megabytes per second, recorded at these times t in seconds. With the trapezoidal rule, estimate how much of it downloads between t=0 and t=4.
    t01234r61081218
  3. Rain falls at a rate r in millimeters per hour, measured at these times t in hours. What estimate of how much rain falls does the trapezoidal rule give?
    t01234r2018201920

  1. Use a midpoint sum from a table

    A car's speed v in feet per second is recorded below. Use a midpoint sum with 3 equal subintervals to estimate the distance it travels from t=0 to t=12.t024681012v(t)306135131121
  2. A car's speed v in feet per second is recorded below. Use a midpoint sum with 3 equal subintervals to estimate the distance it travels from t=0 to t=12.t024681012v(t)246234205
  3. A car's speed v in feet per second is recorded below. Use a midpoint sum with 3 equal subintervals to estimate the distance it travels from t=0 to t=12.t024681012v(t)1727261821415

  1. Estimate average value from a table

    Use a trapezoidal sum to approximate the average value of R on [0,8]. t0368R(t)34351410
  2. Use a trapezoidal sum to approximate the average value of R on [0,7]. t0247R(t)3437129
  3. Use a trapezoidal sum to approximate the average value of R on [0,8]. t0258R(t)40283830

  1. Approximate an integral with negative values

    Use a left Riemann sum with the 4 subintervalsof the table to approximate ∫04f(x) dx.x01234f(x)7−4−731
  2. Use a trapezoidal Riemann sum with the 4 subintervalsof the table to approximate ∫08f(x) dx.x02468f(x)1068−3
  3. Use a left Riemann sum with the 4 subintervalsof the table to approximate ∫012f(x) dx.x036912f(x)−6−8−17−2

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Riemann Sums From a TableAnswer keyVersion 1

Date Period

Estimate each value from the table.

  1. Use a left sum from a table

    Oil leaks from a tank at a rate r in gallons per hour, measured at these times t in hours. Use a left Riemann sum to estimate how much oil leaks out from t=0 to t=8.
    t02468r1218242224    152 gal
  2. Rain falls at a rate r in millimeters per hour, measured at these times t in hours. With a left Riemann sum, estimate how much rain falls between t=0 and t=20.
    t05101520r1621171520    345 mm
  3. A runner's speed v in m/s is measured at these times t in seconds. Estimate how far he runs, using a left Riemann sum.
    t02468v46969    50 m
  4. A car's speed v in m/s is recorded at these times t in seconds. What estimate of how far it travels does a left Riemann sum give?
    t02468v711121311    86 m

  1. Use a right sum from a table

    Water flows into a tank at a rate r in liters per minute, recorded at these times t in minutes. What estimate of how much water flows in does a right Riemann sum give?
    t0510152025r722486    110 L
  2. A car's speed v in m/s is recorded at these times t in seconds. Use a right Riemann sum to estimate how far it travels from t=0 to t=10.
    t0246810v161925202624    228 m
  3. A cyclist's speed v in ft/s is recorded at these times t in seconds. Estimate how far she rides, using a right Riemann sum.
    t51015202530v77612129    230 ft
  4. Water flows into a tank at a rate r in liters per minute, recorded at these times t in minutes. With a right Riemann sum, estimate how much water flows in between t=5 and t=30.
    t51015202530r121318161116    370 L

  1. Use the trapezoidal rule from a table

    A house draws electric power P in kilowatts, read at these times t in hours. Estimate how much energy it uses, using the trapezoidal rule.
    t12345P1218211720    72 kWh
  2. A file downloads at a rate r in megabytes per second, recorded at these times t in seconds. With the trapezoidal rule, estimate how much of it downloads between t=0 and t=4.
    t01234r61081218    42 MB
  3. Rain falls at a rate r in millimeters per hour, measured at these times t in hours. What estimate of how much rain falls does the trapezoidal rule give?
    t01234r2018201920    77 mm

  1. Use a midpoint sum from a table

    A car's speed v in feet per second is recorded below. Use a midpoint sum with 3 equal subintervals to estimate the distance it travels from t=0 to t=12.t024681012v(t)30613513112188 feet
  2. A car's speed v in feet per second is recorded below. Use a midpoint sum with 3 equal subintervals to estimate the distance it travels from t=0 to t=12.t024681012v(t)246234205116 feet
  3. A car's speed v in feet per second is recorded below. Use a midpoint sum with 3 equal subintervals to estimate the distance it travels from t=0 to t=12.t024681012v(t)1727261821415236 feet

  1. Estimate average value from a table

    Use a trapezoidal sum to approximate the average value of R on [0,8]. t0368R(t)343514102018
  2. Use a trapezoidal sum to approximate the average value of R on [0,7]. t0247R(t)343712930314
  3. Use a trapezoidal sum to approximate the average value of R on [0,8]. t0258R(t)402838302698

  1. Approximate an integral with negative values

    Use a left Riemann sum with the 4 subintervalsof the table to approximate ∫04f(x) dx.x01234f(x)7−4−731−1
  2. Use a trapezoidal Riemann sum with the 4 subintervalsof the table to approximate ∫08f(x) dx.x02468f(x)1068−326
  3. Use a left Riemann sum with the 4 subintervalsof the table to approximate ∫012f(x) dx.x036912f(x)−6−8−17−2−24

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Speeds at t=0,2,4 are 5,9,7 m/s. Estimate the distance with a left sum.

  1. Each strip is 2 seconds wide.
  2. A left sum uses the readings at the start of each strip, so 5 and 9.
  3. That gives 2 times 5 plus 2 times 9, which is 28 meters.

The answer is 28 m.

Where students go wrong

Using one width for every strip when the readings are unevenly spaced. Each reading is multiplied by the width of its own strip; the width can be factored out only when every strip is the same, and only then does the trapezoidal rule halve just the two end readings.

Also called Riemann sum table, estimate from a table, trapezoidal rule from a table or distance from a velocity table.

What you can put on this worksheet

Left Riemann sum
A house draws electric power P in kilowatts, read at these times t in hours. Use a left Riemann sum to estimate how much energy it uses from t=0 to t=8.
t02468P161111129 → 100 kWh
Right Riemann sum
A house draws electric power P in kilowatts, read at these times t in hours. Use a right Riemann sum to estimate how much energy it uses from t=0 to t=8.
t02468P161111129 → 86 kWh
The trapezoidal rule
A house draws electric power P in kilowatts, read at these times t in hours. Use the trapezoidal rule to estimate how much energy it uses from t=0 to t=8.
t02468P161111129 → 93 kWh
Midpoint sums
A car's speed v in feet per second is recorded below. Use a midpoint sum with 2 equal subintervals to estimate the distance it travels from t=0 to t=8.t02468v(t)171823415 → 88 feet
Average value from a table
Use a trapezoidal sum to approximate the average value of R on [0,10]. t03610R(t)7222513 → 19
Negative readings (a signed sum)
Use a left Riemann sum with the 4 subintervalsof the table to approximate ∫08f(x) dx.x02468f(x)14−801 → −6

Questions about these worksheets

Yes. Every riemann sums from a table sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the riemann sums from a table questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: left riemann sum, right riemann sum, the trapezoidal rule, midpoint sums, average value from a table and negative readings (a signed sum). Tick as many as you want and set how many of each, or let it spread them evenly.

Using one width for every strip when the readings are unevenly spaced. Each reading is multiplied by the width of its own strip; the width can be factored out only when every strip is the same, and only then does the trapezoidal rule halve just the two end readings.