Precalculus area under a curve by limits worksheet

In Precalculus this is the first look at area as a limit: estimate with rectangles, write the total with sigma notation, then use the summation formulas and let n grow. It is the bridge from sums and limits to the integral that Calculus names.

Also taught in Calculus, at its own level and with its own examples.

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Sigma Prep · sigmaprep.io/worksheets Name Area by the Limit Definition Date Period Find each area using the limit of a Riemann sum. Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 1$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2} + 5) d x$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 4$ Use the limit of a right Riemann sum to evaluate $\int_{1}^{3} (3 x^{2} + 3) d x$ Use the limit of a Riemann sum to find the area under $y = 2 x^{2}$ from $x = 0$ to $x = 2$ Use the limit of a right Riemann sum to evaluate $\int_{1}^{2} (3 x^{2}) d x$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 7$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (x^{2}) d x$ Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 2$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (3 x^{2} + 3) d x$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 8$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{3} (3 x^{2}) d x$ Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 3$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2} + 4) d x$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 3$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2}) d x$ Use the limit of a Riemann sum to find the area under $y = 6 x^{2}$ from $x = 0$ to $x = 5$ Use the limit of a right Riemann sum to evaluate $\int_{3}^{4} (2 x^{2} + 4) d x$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 5$ Use the limit of a right Riemann sum to evaluate $\int_{3}^{4} (2 x^{2}) d x$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Area by the Limit Definition

Date Period

Find each area using the limit of a Riemann sum.

Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 1$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2} + 5) d x$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 4$
Use the limit of a right Riemann sum to evaluate $\int_{1}^{3} (3 x^{2} + 3) d x$
Use the limit of a Riemann sum to find the area under $y = 2 x^{2}$ from $x = 0$ to $x = 2$
Use the limit of a right Riemann sum to evaluate $\int_{1}^{2} (3 x^{2}) d x$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 7$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (x^{2}) d x$
Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 2$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (3 x^{2} + 3) d x$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 8$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{3} (3 x^{2}) d x$
Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 3$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2} + 4) d x$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 3$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2}) d x$
Use the limit of a Riemann sum to find the area under $y = 6 x^{2}$ from $x = 0$ to $x = 5$
Use the limit of a right Riemann sum to evaluate $\int_{3}^{4} (2 x^{2} + 4) d x$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 5$
Use the limit of a right Riemann sum to evaluate $\int_{3}^{4} (2 x^{2}) d x$



Sigma Prep · sigmaprep.io/worksheets Name Area by the Limit DefinitionAnswer keyVersion 1 Date Period Find each area using the limit of a Riemann sum. Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 1$ $\frac{4}{3}$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2} + 5) d x$ $54$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 4$ $\frac{64}{3}$ Use the limit of a right Riemann sum to evaluate $\int_{1}^{3} (3 x^{2} + 3) d x$ $32$ Use the limit of a Riemann sum to find the area under $y = 2 x^{2}$ from $x = 0$ to $x = 2$ $\frac{16}{3}$ Use the limit of a right Riemann sum to evaluate $\int_{1}^{2} (3 x^{2}) d x$ $7$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 7$ $\frac{343}{3}$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (x^{2}) d x$ $\frac{56}{3}$ Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 2$ $\frac{32}{3}$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (3 x^{2} + 3) d x$ $62$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 8$ $\frac{512}{3}$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{3} (3 x^{2}) d x$ $19$ Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 3$ $36$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2} + 4) d x$ $51$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 3$ $9$ Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2}) d x$ $39$ Use the limit of a Riemann sum to find the area under $y = 6 x^{2}$ from $x = 0$ to $x = 5$ $250$ Use the limit of a right Riemann sum to evaluate $\int_{3}^{4} (2 x^{2} + 4) d x$ $\frac{86}{3}$ Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 5$ $\frac{125}{3}$ Use the limit of a right Riemann sum to evaluate $\int_{3}^{4} (2 x^{2}) d x$ $\frac{74}{3}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Area by the Limit DefinitionAnswer keyVersion 1

Date Period

Find each area using the limit of a Riemann sum.

Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 1$ $\frac{4}{3}$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2} + 5) d x$ $54$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 4$ $\frac{64}{3}$
Use the limit of a right Riemann sum to evaluate $\int_{1}^{3} (3 x^{2} + 3) d x$ $32$
Use the limit of a Riemann sum to find the area under $y = 2 x^{2}$ from $x = 0$ to $x = 2$ $\frac{16}{3}$
Use the limit of a right Riemann sum to evaluate $\int_{1}^{2} (3 x^{2}) d x$ $7$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 7$ $\frac{343}{3}$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (x^{2}) d x$ $\frac{56}{3}$
Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 2$ $\frac{32}{3}$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (3 x^{2} + 3) d x$ $62$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 8$ $\frac{512}{3}$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{3} (3 x^{2}) d x$ $19$
Use the limit of a Riemann sum to find the area under $y = 4 x^{2}$ from $x = 0$ to $x = 3$ $36$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2} + 4) d x$ $51$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 3$ $9$
Use the limit of a right Riemann sum to evaluate $\int_{2}^{5} (x^{2}) d x$ $39$
Use the limit of a Riemann sum to find the area under $y = 6 x^{2}$ from $x = 0$ to $x = 5$ $250$
Use the limit of a right Riemann sum to evaluate $\int_{3}^{4} (2 x^{2} + 4) d x$ $\frac{86}{3}$
Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 5$ $\frac{125}{3}$
Use the limit of a right Riemann sum to evaluate $\int_{3}^{4} (2 x^{2}) d x$ $\frac{74}{3}$

How to do these

Use the limit of a Riemann sum to find the area under $y = x^{2}$ from $x = 0$ to $x = 3$

Each strip is 3/n wide and the kth right endpoint is 3k/n, so the kth rectangle is (9k²/n²)(3/n).
The sum is 27/n³ times the sum of k², which is n(n + 1)(2n + 1)/6.
That is 27(n + 1)(2n + 1)/(6n²), which tends to 27 × 2/6 as n grows.

The answer is $9$ .

Where students go wrong

Using the formula for the sum of k when the terms hold k². Squared heights need n(n + 1)(2n + 1)/6, not n(n + 1)/2.

Also called limit definition of the integral, area as a limit, Riemann sum limit or summation formulas.

What you can put on this worksheet

Area as a limit of sums: Use the limit of a Riemann sum to find the area under $y = 4 x$ from $x = 0$ to $x = 5$ → $50$
Intervals that do not start at 0: Use the limit of a right Riemann sum to evaluate $\int_{2}^{4} (x + 4) d x$ → $14$
A sum in closed form: Write in closed form: $\sum_{k = 1}^{n} (3 k + 3)$ → $\frac{3}{2} n^{2} + \frac{9}{2} n$

Questions about these worksheets

Yes. Every area under a curve by limits 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 area under a curve by limits 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 3: area as a limit of sums, intervals that do not start at 0 and a sum in closed form. Tick as many as you want and set how many of each, or let it spread them evenly.

Letting n go to infinity before summing. The summation formula has to be used first, and only then does the n in it cancel with the n from the width.

Precalculus area under a curve by limits worksheet

Select your difficulty

Area by the Limit Definition

Area by the Limit DefinitionAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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