In Precalculus Riemann sums are the first look at area under a curve as a limit. You add up rectangles, see the estimate improve as the strips get thinner, and write the sum in sigma notation.
Also taught in Calculus, at its own level and with its own examples.
Select your difficulty
Pick more than one for a sheet that mixes them.
The preview can look cramped on a phone. The PDF and printed sheet come out normal.
Right Riemann sum for f(x)=x2+1on [0,2] with 4 strips
Each strip is 2/4 = 0.5 wide, and the right edges are 0.5, 1, 1.5 and 2.
The heights are 1.25, 2, 3.25 and 5, which add to 11.5.
Multiply by the width: 11.5 × 0.5.
The answer is 5.75.
Where students go wrong
Adding the heights and stopping. Each rectangle is height times width, so the sum of heights has to be multiplied by the strip width.
Also called left Riemann sum, right Riemann sum, midpoint rule, trapezoidal rule or approximating area under a curve.
What you can put on this worksheet
Left endpoints
Estimate the area under y=3x+3from x=0to x=4 using 4 strips and left endpoints.→ 30
Right endpoints
Estimate the area under y=3x+3from x=0to x=4 using 4 strips and right endpoints.→ 42
Midpoints
Estimate the area under y=3x+3from x=0to x=4 using 4 strips and midpoints.→ 36
The trapezoidal rule
Estimate the area under y=3x+3from x=0to x=4 using 4 strips and the trapezoidal rule.→ 36
Read off a graph
Use a left Riemann sum with 4 equal subintervals to approximate ∫−26f(x)dx— the graph of f on [-2, 6]→ 12
The limit of a sum, as an integral
n→∞lim∑i=1n(2+n2i)2⋅n2→ 356
Over- or underestimate?
f is concave up on [0,4]. Is the trapezoidal sum an overestimate or an underestimate of ∫04f(x)dx?→ Overestimate
Sketch the rectangles yourself
Sketch f(x)=2xon [1,5], draw 4 left-endpoint rectangles, and approximate the area to the nearest hundredth.→ 12.29
Questions about these worksheets
Yes. Every riemann sums 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.
Precalculus is usually taken around 11th or 12th grade, 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 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 three 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 8: left endpoints, right endpoints, midpoints, the trapezoidal rule, read off a graph, the limit of a sum, as an integral, over- or underestimate? and sketch the rectangles yourself. Tick as many as you want and set how many of each, or let it spread them evenly.
Using one endpoint too many, or one too few. Four strips have four left edges and four right edges, but five edges in total, so the left sum skips the last one and the right sum skips the first.