Find S_4 and the exact value of integral of_0^4 (-3x^4 - x^3 - 2x) dx. What is the error?

S_4 = -696; Exact value: -3472/5; Error: 8/5

Find S_4 and the exact value of integral of_0^4 (-x^4 + 3x^3 - x + 4) dx. What is the error?

S_4 = -16/3; Exact value: -24/5; Error: 8/15

Find S_4 and the exact value of integral of_-1^1 (2x^4 - 3x^3 - 2x + 4) dx. What is the error?

S_4 = 53/6; Exact value: 44/5; Error: 1/30

Find S_4 and the exact value of integral of_-1^1 (x^4 + 3x^3 + x + 4) dx. What is the error?

S_4 = 101/12; Exact value: 42/5; Error: 1/60

Find S_4 and the exact value of integral of_-1^3 -2x^4 dx. What is the error?

S_4 = -296/3; Exact value: -488/5; Error: 16/15

Find S_4 and the exact value of integral of_1^3 (-3x^4 + x^3 - 2x + 2) dx. What is the error?

S_4 = -517/4; Exact value: -646/5; Error: 1/20

Find S_2 and the exact value of integral of_1^3 (2x^3 + 2x^2) dx. What is the error?

S_2 = 172/3; Exact value: 172/3; Error: 0

Worksheets · Calculus

Simpson's rule worksheet

Simpson's rule fits a parabola through each pair of strips, so n has to be even. The heights are weighted 1, 4, 2, 4, ..., 2, 4, 1 and the total is multiplied by h/3. It is exact for any cubic.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Simpson's Rule Date Period Use Simpson's rule to answer each question. Use Simpson's rule with $n = 4$ to approximate $\int_{0}^{4} (- 2 x^{4} + 3 x^{3} + 4 x^{2} + 1) d x$ . Use Simpson's rule with the table to approximate $\int_{1}^{13} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 4 & 7 & 10 & 13 \\ f (x) & 5 & 1 & - 1 & 7 & - 4 \end{array}$ Find $S_{4}$ and the exact value of $\int_{0}^{4} (- 3 x^{4} - x^{3} - 2 x) d x$ . What is the error? Use Simpson's rule with $n = 4$ to approximate $\int_{2}^{6} \frac{4}{x^{2}} d x$ . Oil leaks from a tank at the rates shown. Use Simpson's rule to estimate how much leaks from $t = 2$ to $t = 10$ : $\begin{array}{cccccc} t (h) & 2 & 4 & 6 & 8 & 10 \\ r (t) (gal/h) & 19 & 14 & 19 & 17 & 19 \end{array}$ Find $S_{4}$ and the exact value of $\int_{0}^{4} (- x^{4} + 3 x^{3} - x + 4) d x$ . What is the error? Use Simpson's rule with $n = 4$ to approximate $\int_{0}^{4} \frac{12}{x + 2} d x$ . A runner's velocity is read at the times shown. Use Simpson's rule to estimate the distance run from $t = 1$ to $t = 9$ : $\begin{array}{cccccc} t (s) & 1 & 3 & 5 & 7 & 9 \\ v (t) (m/s) & 7 & 12 & 0 & 19 & 11 \end{array}$ Find $S_{4}$ and the exact value of $\int_{- 1}^{1} (2 x^{4} - 3 x^{3} - 2 x + 4) d x$ . What is the error? Use Simpson's rule with $n = 4$ to approximate $\int_{- 2}^{2} (- 2 x^{4} + 3 x^{3} - 4 x^{2}) d x$ . Use Simpson's rule with the table to approximate $\int_{2}^{10} f (x) d x$ : $\begin{array}{cccccc} x & 2 & 4 & 6 & 8 & 10 \\ f (x) & - 3 & 20 & 20 & - 5 & 17 \end{array}$ Find $S_{4}$ and the exact value of $\int_{- 1}^{1} (x^{4} + 3 x^{3} + x + 4) d x$ . What is the error? Use Simpson's rule with $n = 4$ to approximate $\int_{3}^{7} \frac{2}{x} d x$ . Oil leaks from a tank at the rates shown. Use Simpson's rule to estimate how much leaks from $t = 0$ to $t = 12$ : $\begin{array}{cccccc} t (h) & 0 & 3 & 6 & 9 & 12 \\ r (t) (gal/h) & 2 & 4 & 11 & 11 & 12 \end{array}$ Find $S_{4}$ and the exact value of $\int_{- 1}^{3} - 2 x^{4} d x$ . What is the error? Use Simpson's rule with $n = 4$ to approximate $\int_{- 2}^{0} (x^{4} + x^{3} - 3 x^{2} + 5) d x$ . A runner's velocity is read at the times shown. Use Simpson's rule to estimate the distance run from $t = 2$ to $t = 14$ : $\begin{array}{cccccc} t (s) & 2 & 5 & 8 & 11 & 14 \\ v (t) (m/s) & 6 & 4 & 13 & 8 & 17 \end{array}$ Find $S_{4}$ and the exact value of $\int_{1}^{3} (- 3 x^{4} + x^{3} - 2 x + 2) d x$ . What is the error? Use Simpson's rule with $n = 4$ to approximate $\int_{- 1}^{3} (- 2 x^{4} + x^{3} + 2 x^{2} + 2) d x$ . Use Simpson's rule with the table to approximate $\int_{1}^{5} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ f (x) & 17 & 0 & 9 & 1 & 15 \end{array}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Simpson's Rule

Date Period

Use Simpson's rule to answer each question.

Use Simpson's rule with $n = 4$ to approximate $\int_{0}^{4} (- 2 x^{4} + 3 x^{3} + 4 x^{2} + 1) d x$ .
Use Simpson's rule with the table to approximate $\int_{1}^{13} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 4 & 7 & 10 & 13 \\ f (x) & 5 & 1 & - 1 & 7 & - 4 \end{array}$
Find $S_{4}$ and the exact value of $\int_{0}^{4} (- 3 x^{4} - x^{3} - 2 x) d x$ . What is the error?
Use Simpson's rule with $n = 4$ to approximate $\int_{2}^{6} \frac{4}{x^{2}} d x$ .
Oil leaks from a tank at the rates shown. Use Simpson's rule to estimate how much leaks from $t = 2$ to $t = 10$ : $\begin{array}{cccccc} t (h) & 2 & 4 & 6 & 8 & 10 \\ r (t) (gal/h) & 19 & 14 & 19 & 17 & 19 \end{array}$
Find $S_{4}$ and the exact value of $\int_{0}^{4} (- x^{4} + 3 x^{3} - x + 4) d x$ . What is the error?
Use Simpson's rule with $n = 4$ to approximate $\int_{0}^{4} \frac{12}{x + 2} d x$ .
A runner's velocity is read at the times shown. Use Simpson's rule to estimate the distance run from $t = 1$ to $t = 9$ : $\begin{array}{cccccc} t (s) & 1 & 3 & 5 & 7 & 9 \\ v (t) (m/s) & 7 & 12 & 0 & 19 & 11 \end{array}$
Find $S_{4}$ and the exact value of $\int_{- 1}^{1} (2 x^{4} - 3 x^{3} - 2 x + 4) d x$ . What is the error?
Use Simpson's rule with $n = 4$ to approximate $\int_{- 2}^{2} (- 2 x^{4} + 3 x^{3} - 4 x^{2}) d x$ .
Use Simpson's rule with the table to approximate $\int_{2}^{10} f (x) d x$ : $\begin{array}{cccccc} x & 2 & 4 & 6 & 8 & 10 \\ f (x) & - 3 & 20 & 20 & - 5 & 17 \end{array}$
Find $S_{4}$ and the exact value of $\int_{- 1}^{1} (x^{4} + 3 x^{3} + x + 4) d x$ . What is the error?
Use Simpson's rule with $n = 4$ to approximate $\int_{3}^{7} \frac{2}{x} d x$ .
Oil leaks from a tank at the rates shown. Use Simpson's rule to estimate how much leaks from $t = 0$ to $t = 12$ : $\begin{array}{cccccc} t (h) & 0 & 3 & 6 & 9 & 12 \\ r (t) (gal/h) & 2 & 4 & 11 & 11 & 12 \end{array}$
Find $S_{4}$ and the exact value of $\int_{- 1}^{3} - 2 x^{4} d x$ . What is the error?
Use Simpson's rule with $n = 4$ to approximate $\int_{- 2}^{0} (x^{4} + x^{3} - 3 x^{2} + 5) d x$ .
A runner's velocity is read at the times shown. Use Simpson's rule to estimate the distance run from $t = 2$ to $t = 14$ : $\begin{array}{cccccc} t (s) & 2 & 5 & 8 & 11 & 14 \\ v (t) (m/s) & 6 & 4 & 13 & 8 & 17 \end{array}$
Find $S_{4}$ and the exact value of $\int_{1}^{3} (- 3 x^{4} + x^{3} - 2 x + 2) d x$ . What is the error?
Use Simpson's rule with $n = 4$ to approximate $\int_{- 1}^{3} (- 2 x^{4} + x^{3} + 2 x^{2} + 2) d x$ .
Use Simpson's rule with the table to approximate $\int_{1}^{5} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ f (x) & 17 & 0 & 9 & 1 & 15 \end{array}$



Sigma Prep · sigmaprep.io/worksheets Name Simpson's RuleAnswer keyVersion 1 Date Period Use Simpson's rule to answer each question. Use Simpson's rule with $n = 4$ to approximate $\int_{0}^{4} (- 2 x^{4} + 3 x^{3} + 4 x^{2} + 1) d x$ . $- \frac{388}{3}$ Use Simpson's rule with the table to approximate $\int_{1}^{13} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 4 & 7 & 10 & 13 \\ f (x) & 5 & 1 & - 1 & 7 & - 4 \end{array}$ $31$ Find $S_{4}$ and the exact value of $\int_{0}^{4} (- 3 x^{4} - x^{3} - 2 x) d x$ . What is the error? $\begin{array}{l} S_{4} = - 696 \\ Exact value: - \frac{3472}{5} \\ Error: \frac{8}{5} \end{array}$ Use Simpson's rule with $n = 4$ to approximate $\int_{2}^{6} \frac{4}{x^{2}} d x$ . $\frac{1813}{1350}$ Oil leaks from a tank at the rates shown. Use Simpson's rule to estimate how much leaks from $t = 2$ to $t = 10$ : $\begin{array}{cccccc} t (h) & 2 & 4 & 6 & 8 & 10 \\ r (t) (gal/h) & 19 & 14 & 19 & 17 & 19 \end{array}$ $\frac{400}{3}$ gallons Find $S_{4}$ and the exact value of $\int_{0}^{4} (- x^{4} + 3 x^{3} - x + 4) d x$ . What is the error? $\begin{array}{l} S_{4} = - \frac{16}{3} \\ Exact value: - \frac{24}{5} \\ Error: \frac{8}{15} \end{array}$ Use Simpson's rule with $n = 4$ to approximate $\int_{0}^{4} \frac{12}{x + 2} d x$ . $\frac{66}{5}$ A runner's velocity is read at the times shown. Use Simpson's rule to estimate the distance run from $t = 1$ to $t = 9$ : $\begin{array}{cccccc} t (s) & 1 & 3 & 5 & 7 & 9 \\ v (t) (m/s) & 7 & 12 & 0 & 19 & 11 \end{array}$ $\frac{284}{3}$ meters Find $S_{4}$ and the exact value of $\int_{- 1}^{1} (2 x^{4} - 3 x^{3} - 2 x + 4) d x$ . What is the error? $\begin{array}{l} S_{4} = \frac{53}{6} \\ Exact value: \frac{44}{5} \\ Error: \frac{1}{30} \end{array}$ Use Simpson's rule with $n = 4$ to approximate $\int_{- 2}^{2} (- 2 x^{4} + 3 x^{3} - 4 x^{2}) d x$ . $- 48$ Use Simpson's rule with the table to approximate $\int_{2}^{10} f (x) d x$ : $\begin{array}{cccccc} x & 2 & 4 & 6 & 8 & 10 \\ f (x) & - 3 & 20 & 20 & - 5 & 17 \end{array}$ $76$ Find $S_{4}$ and the exact value of $\int_{- 1}^{1} (x^{4} + 3 x^{3} + x + 4) d x$ . What is the error? $\begin{array}{l} S_{4} = \frac{101}{12} \\ Exact value: \frac{42}{5} \\ Error: \frac{1}{60} \end{array}$ Use Simpson's rule with $n = 4$ to approximate $\int_{3}^{7} \frac{2}{x} d x$ . $\frac{178}{105}$ Oil leaks from a tank at the rates shown. Use Simpson's rule to estimate how much leaks from $t = 0$ to $t = 12$ : $\begin{array}{cccccc} t (h) & 0 & 3 & 6 & 9 & 12 \\ r (t) (gal/h) & 2 & 4 & 11 & 11 & 12 \end{array}$ $96$ gallons Find $S_{4}$ and the exact value of $\int_{- 1}^{3} - 2 x^{4} d x$ . What is the error? $\begin{array}{l} S_{4} = - \frac{296}{3} \\ Exact value: - \frac{488}{5} \\ Error: \frac{16}{15} \end{array}$ Use Simpson's rule with $n = 4$ to approximate $\int_{- 2}^{0} (x^{4} + x^{3} - 3 x^{2} + 5) d x$ . $\frac{53}{12}$ A runner's velocity is read at the times shown. Use Simpson's rule to estimate the distance run from $t = 2$ to $t = 14$ : $\begin{array}{cccccc} t (s) & 2 & 5 & 8 & 11 & 14 \\ v (t) (m/s) & 6 & 4 & 13 & 8 & 17 \end{array}$ $97$ meters Find $S_{4}$ and the exact value of $\int_{1}^{3} (- 3 x^{4} + x^{3} - 2 x + 2) d x$ . What is the error? $\begin{array}{l} S_{4} = - \frac{517}{4} \\ Exact value: - \frac{646}{5} \\ Error: \frac{1}{20} \end{array}$ Use Simpson's rule with $n = 4$ to approximate $\int_{- 1}^{3} (- 2 x^{4} + x^{3} + 2 x^{2} + 2) d x$ . $- 52$ Use Simpson's rule with the table to approximate $\int_{1}^{5} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ f (x) & 17 & 0 & 9 & 1 & 15 \end{array}$ $18$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Simpson's RuleAnswer keyVersion 1

Date Period

Use Simpson's rule to answer each question.

Use Simpson's rule with $n = 4$ to approximate $\int_{0}^{4} (- 2 x^{4} + 3 x^{3} + 4 x^{2} + 1) d x$ . $- \frac{388}{3}$
Use Simpson's rule with the table to approximate $\int_{1}^{13} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 4 & 7 & 10 & 13 \\ f (x) & 5 & 1 & - 1 & 7 & - 4 \end{array}$ $31$
Find $S_{4}$ and the exact value of $\int_{0}^{4} (- 3 x^{4} - x^{3} - 2 x) d x$ . What is the error? $\begin{array}{l} S_{4} = - 696 \\ Exact value: - \frac{3472}{5} \\ Error: \frac{8}{5} \end{array}$
Use Simpson's rule with $n = 4$ to approximate $\int_{2}^{6} \frac{4}{x^{2}} d x$ . $\frac{1813}{1350}$
Oil leaks from a tank at the rates shown. Use Simpson's rule to estimate how much leaks from $t = 2$ to $t = 10$ : $\begin{array}{cccccc} t (h) & 2 & 4 & 6 & 8 & 10 \\ r (t) (gal/h) & 19 & 14 & 19 & 17 & 19 \end{array}$ $\frac{400}{3}$ gallons
Find $S_{4}$ and the exact value of $\int_{0}^{4} (- x^{4} + 3 x^{3} - x + 4) d x$ . What is the error? $\begin{array}{l} S_{4} = - \frac{16}{3} \\ Exact value: - \frac{24}{5} \\ Error: \frac{8}{15} \end{array}$
Use Simpson's rule with $n = 4$ to approximate $\int_{0}^{4} \frac{12}{x + 2} d x$ . $\frac{66}{5}$
A runner's velocity is read at the times shown. Use Simpson's rule to estimate the distance run from $t = 1$ to $t = 9$ : $\begin{array}{cccccc} t (s) & 1 & 3 & 5 & 7 & 9 \\ v (t) (m/s) & 7 & 12 & 0 & 19 & 11 \end{array}$ $\frac{284}{3}$ meters
Find $S_{4}$ and the exact value of $\int_{- 1}^{1} (2 x^{4} - 3 x^{3} - 2 x + 4) d x$ . What is the error? $\begin{array}{l} S_{4} = \frac{53}{6} \\ Exact value: \frac{44}{5} \\ Error: \frac{1}{30} \end{array}$
Use Simpson's rule with $n = 4$ to approximate $\int_{- 2}^{2} (- 2 x^{4} + 3 x^{3} - 4 x^{2}) d x$ . $- 48$
Use Simpson's rule with the table to approximate $\int_{2}^{10} f (x) d x$ : $\begin{array}{cccccc} x & 2 & 4 & 6 & 8 & 10 \\ f (x) & - 3 & 20 & 20 & - 5 & 17 \end{array}$ $76$
Find $S_{4}$ and the exact value of $\int_{- 1}^{1} (x^{4} + 3 x^{3} + x + 4) d x$ . What is the error? $\begin{array}{l} S_{4} = \frac{101}{12} \\ Exact value: \frac{42}{5} \\ Error: \frac{1}{60} \end{array}$
Use Simpson's rule with $n = 4$ to approximate $\int_{3}^{7} \frac{2}{x} d x$ . $\frac{178}{105}$
Oil leaks from a tank at the rates shown. Use Simpson's rule to estimate how much leaks from $t = 0$ to $t = 12$ : $\begin{array}{cccccc} t (h) & 0 & 3 & 6 & 9 & 12 \\ r (t) (gal/h) & 2 & 4 & 11 & 11 & 12 \end{array}$ $96$ gallons
Find $S_{4}$ and the exact value of $\int_{- 1}^{3} - 2 x^{4} d x$ . What is the error? $\begin{array}{l} S_{4} = - \frac{296}{3} \\ Exact value: - \frac{488}{5} \\ Error: \frac{16}{15} \end{array}$
Use Simpson's rule with $n = 4$ to approximate $\int_{- 2}^{0} (x^{4} + x^{3} - 3 x^{2} + 5) d x$ . $\frac{53}{12}$
A runner's velocity is read at the times shown. Use Simpson's rule to estimate the distance run from $t = 2$ to $t = 14$ : $\begin{array}{cccccc} t (s) & 2 & 5 & 8 & 11 & 14 \\ v (t) (m/s) & 6 & 4 & 13 & 8 & 17 \end{array}$ $97$ meters
Find $S_{4}$ and the exact value of $\int_{1}^{3} (- 3 x^{4} + x^{3} - 2 x + 2) d x$ . What is the error? $\begin{array}{l} S_{4} = - \frac{517}{4} \\ Exact value: - \frac{646}{5} \\ Error: \frac{1}{20} \end{array}$
Use Simpson's rule with $n = 4$ to approximate $\int_{- 1}^{3} (- 2 x^{4} + x^{3} + 2 x^{2} + 2) d x$ . $- 52$
Use Simpson's rule with the table to approximate $\int_{1}^{5} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ f (x) & 17 & 0 & 9 & 1 & 15 \end{array}$ $18$

How to do these

Use Simpson's rule with $n = 2$ to approximate $\int_{0}^{2} x^{3} d x$ .

h = 1, and the heights are f(0) = 0, f(1) = 1 and f(2) = 8.
S_2 = (1/3)(0 + 4 · 1 + 8) = 4.
The exact value is also 4: Simpson is exact for cubics.

The answer is $4$ .

Where students go wrong

Using an odd number of subintervals, or weighting every inside height by 4. The pattern alternates 4, 2, 4, 2 and needs n even to finish on a 4.

Also called Simpson's 1/3 rule, Simpson rule, parabolic rule or Simpson's approximation.

What you can put on this worksheet

Simpson's rule from a function: Use Simpson's rule with $n = 2$ to approximate $\int_{1}^{5} (x^{2} + x + 3) d x$ . → $\frac{196}{3}$
Simpson's rule from a table: Use Simpson's rule with the table to approximate $\int_{0}^{4} f (x) d x$ : $\begin{array}{cccc} x & 0 & 2 & 4 \\ f (x) & 9 & 1 & 20 \end{array}$ → $22$
Against the exact value: Find $S_{2}$ and the exact value of $\int_{1}^{3} (2 x^{3} + 2 x^{2}) d x$ . What is the error? → $\begin{array}{l} S_{2} = \frac{172}{3} \\ Exact value: \frac{172}{3} \\ Error: 0 \end{array}$

Questions about these worksheets

Yes. Every simpson's rule 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 simpson's rule 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: simpson's rule from a function, simpson's rule from a table and against the exact value. Tick as many as you want and set how many of each, or let it spread them evenly.

Using an odd number of subintervals, or weighting every inside height by 4. The pattern alternates 4, 2, 4, 2 and needs n even to finish on a 4.

Simpson's rule worksheet

Select your difficulty

Simpson's Rule

Simpson's RuleAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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