Does the trapezoidal rule overestimate or underestimate integral of_1^4 1/x dx? Why?

Overestimate: the curve is concave up

Trapezoidal rule worksheet

Each trapezoid's area is its width times the average of its two heights. On a curve that bends up the tops sit above the curve, so the rule overestimates; on one that bends down it underestimates.

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Sigma Prep · sigmaprep.io/worksheets Name The Trapezoidal Rule Date Period Use the trapezoidal rule to answer each question. Estimate the area under $y = 3 x^{2} - 3 x + 4$ from $x = 3$ to $x = 7$ using 4 strips and the trapezoidal rule. A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 2 & 4 & 6 & 8 & 10 & 12 \\ v & 11 & 17 & 17 & 17 & 13 & 12 \end{array}$ Water flows into a pond at the rates shown. Use trapezoids to estimate how much flows in from $t = 0$ to $t = 6$ : $\begin{array}{cccccc} t (h) & 0 & 1 & 3 & 4 & 6 \\ r (t) (gal/h) & 6 & 4 & 0 & 8 & 12 \end{array}$ Find $T_{2}$ for $\int_{0}^{2} \frac{6}{x + 1} d x$ . Is it an overestimate or an underestimate? Bound the error of $T_{8}$ for $\int_{0}^{1} x^{4} d x$ . Estimate the area under $y = 3 x^{2} + x + 3$ from $x = 1$ to $x = 7$ using 6 strips and the trapezoidal rule. A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 5 & 10 & 15 & 20 & 25 & 30 \\ v & 19 & 15 & 10 & 15 & 13 & 18 \end{array}$ A car's velocity is read at the times shown. Use trapezoids to estimate how far it travels from $t = 0$ to $t = 9$ : $\begin{array}{cccccc} t (s) & 0 & 3 & 6 & 8 & 9 \\ v (t) (ft/s) & 11 & 7 & 20 & 13 & 10 \end{array}$ Find $T_{2}$ for $\int_{0}^{2} (- x^{3} + 8) d x$ . Is it an overestimate or an underestimate? Bound the error of $T_{4}$ for $\int_{0}^{3} \sin x d x$ . Estimate the area under $y = x^{2} - 3 x + 3$ from $x = 0$ to $x = 4$ using 4 strips and the trapezoidal rule. A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 0 & 1 & 2 & 3 & 4 & 5 \\ v & 14 & 16 & 11 & 10 & 7 & 11 \end{array}$ Use the trapezoids the table makes to approximate $\int_{1}^{7} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 3 & 4 & 5 & 7 \\ f (x) & - 1 & 1 & 11 & 4 & - 3 \end{array}$ Find $T_{2}$ for $\int_{1}^{3} (x^{3} - 2 x^{2} + 5) d x$ . Is it an overestimate or an underestimate? Bound the error of $T_{6}$ for $\int_{1}^{3} \frac{1}{x} d x$ . Estimate the area under $y = x^{2} + 2 x + 1$ from $x = 2$ to $x = 8$ using 6 strips and the trapezoidal rule. A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 0 & 5 & 10 & 15 & 20 & 25 \\ v & 14 & 12 & 18 & 14 & 9 & 11 \end{array}$ Use the trapezoids the table makes to approximate $\int_{2}^{13} f (x) d x$ : $\begin{array}{cccccc} x & 2 & 5 & 8 & 11 & 13 \\ f (x) & 17 & 20 & 14 & 2 & 1 \end{array}$ Find $T_{2}$ for $\int_{0}^{4} x^{2} d x$ . Is it an overestimate or an underestimate? Bound the error of $T_{5}$ for $\int_{0}^{2} \cos 2 x d x$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Trapezoidal Rule

Date Period

Use the trapezoidal rule to answer each question.

Estimate the area under $y = 3 x^{2} - 3 x + 4$ from $x = 3$ to $x = 7$ using 4 strips and the trapezoidal rule.
A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 2 & 4 & 6 & 8 & 10 & 12 \\ v & 11 & 17 & 17 & 17 & 13 & 12 \end{array}$
Water flows into a pond at the rates shown. Use trapezoids to estimate how much flows in from $t = 0$ to $t = 6$ : $\begin{array}{cccccc} t (h) & 0 & 1 & 3 & 4 & 6 \\ r (t) (gal/h) & 6 & 4 & 0 & 8 & 12 \end{array}$
Find $T_{2}$ for $\int_{0}^{2} \frac{6}{x + 1} d x$ . Is it an overestimate or an underestimate?
Bound the error of $T_{8}$ for $\int_{0}^{1} x^{4} d x$ .
Estimate the area under $y = 3 x^{2} + x + 3$ from $x = 1$ to $x = 7$ using 6 strips and the trapezoidal rule.
A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 5 & 10 & 15 & 20 & 25 & 30 \\ v & 19 & 15 & 10 & 15 & 13 & 18 \end{array}$
A car's velocity is read at the times shown. Use trapezoids to estimate how far it travels from $t = 0$ to $t = 9$ : $\begin{array}{cccccc} t (s) & 0 & 3 & 6 & 8 & 9 \\ v (t) (ft/s) & 11 & 7 & 20 & 13 & 10 \end{array}$
Find $T_{2}$ for $\int_{0}^{2} (- x^{3} + 8) d x$ . Is it an overestimate or an underestimate?
Bound the error of $T_{4}$ for $\int_{0}^{3} \sin x d x$ .
Estimate the area under $y = x^{2} - 3 x + 3$ from $x = 0$ to $x = 4$ using 4 strips and the trapezoidal rule.
A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 0 & 1 & 2 & 3 & 4 & 5 \\ v & 14 & 16 & 11 & 10 & 7 & 11 \end{array}$
Use the trapezoids the table makes to approximate $\int_{1}^{7} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 3 & 4 & 5 & 7 \\ f (x) & - 1 & 1 & 11 & 4 & - 3 \end{array}$
Find $T_{2}$ for $\int_{1}^{3} (x^{3} - 2 x^{2} + 5) d x$ . Is it an overestimate or an underestimate?
Bound the error of $T_{6}$ for $\int_{1}^{3} \frac{1}{x} d x$ .
Estimate the area under $y = x^{2} + 2 x + 1$ from $x = 2$ to $x = 8$ using 6 strips and the trapezoidal rule.
A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 0 & 5 & 10 & 15 & 20 & 25 \\ v & 14 & 12 & 18 & 14 & 9 & 11 \end{array}$
Use the trapezoids the table makes to approximate $\int_{2}^{13} f (x) d x$ : $\begin{array}{cccccc} x & 2 & 5 & 8 & 11 & 13 \\ f (x) & 17 & 20 & 14 & 2 & 1 \end{array}$
Find $T_{2}$ for $\int_{0}^{4} x^{2} d x$ . Is it an overestimate or an underestimate?
Bound the error of $T_{5}$ for $\int_{0}^{2} \cos 2 x d x$ .



Sigma Prep · sigmaprep.io/worksheets Name The Trapezoidal RuleAnswer keyVersion 1 Date Period Use the trapezoidal rule to answer each question. Estimate the area under $y = 3 x^{2} - 3 x + 4$ from $x = 3$ to $x = 7$ using 4 strips and the trapezoidal rule. $274$ A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 2 & 4 & 6 & 8 & 10 & 12 \\ v & 11 & 17 & 17 & 17 & 13 & 12 \end{array}$ $151$ m Water flows into a pond at the rates shown. Use trapezoids to estimate how much flows in from $t = 0$ to $t = 6$ : $\begin{array}{cccccc} t (h) & 0 & 1 & 3 & 4 & 6 \\ r (t) (gal/h) & 6 & 4 & 0 & 8 & 12 \end{array}$ $33$ gallons Find $T_{2}$ for $\int_{0}^{2} \frac{6}{x + 1} d x$ . Is it an overestimate or an underestimate? $\begin{array}{l} T_{2} = 7 \\ An overestimate \end{array}$ Bound the error of $T_{8}$ for $\int_{0}^{1} x^{4} d x$ . $∣ E_{T} ∣ \leq \frac{1}{64}$ Estimate the area under $y = 3 x^{2} + x + 3$ from $x = 1$ to $x = 7$ using 6 strips and the trapezoidal rule. $387$ A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 5 & 10 & 15 & 20 & 25 & 30 \\ v & 19 & 15 & 10 & 15 & 13 & 18 \end{array}$ $\frac{715}{2}$ m A car's velocity is read at the times shown. Use trapezoids to estimate how far it travels from $t = 0$ to $t = 9$ : $\begin{array}{cccccc} t (s) & 0 & 3 & 6 & 8 & 9 \\ v (t) (ft/s) & 11 & 7 & 20 & 13 & 10 \end{array}$ $112$ feet Find $T_{2}$ for $\int_{0}^{2} (- x^{3} + 8) d x$ . Is it an overestimate or an underestimate? $\begin{array}{l} T_{2} = 11 \\ An underestimate \end{array}$ Bound the error of $T_{4}$ for $\int_{0}^{3} \sin x d x$ . $∣ E_{T} ∣ \leq \frac{9}{64}$ Estimate the area under $y = x^{2} - 3 x + 3$ from $x = 0$ to $x = 4$ using 4 strips and the trapezoidal rule. $10$ A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 0 & 1 & 2 & 3 & 4 & 5 \\ v & 14 & 16 & 11 & 10 & 7 & 11 \end{array}$ $\frac{113}{2}$ m Use the trapezoids the table makes to approximate $\int_{1}^{7} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 3 & 4 & 5 & 7 \\ f (x) & - 1 & 1 & 11 & 4 & - 3 \end{array}$ $\frac{29}{2}$ Find $T_{2}$ for $\int_{1}^{3} (x^{3} - 2 x^{2} + 5) d x$ . Is it an overestimate or an underestimate? $\begin{array}{l} T_{2} = 14 \\ An overestimate \end{array}$ Bound the error of $T_{6}$ for $\int_{1}^{3} \frac{1}{x} d x$ . $∣ E_{T} ∣ \leq \frac{1}{27}$ Estimate the area under $y = x^{2} + 2 x + 1$ from $x = 2$ to $x = 8$ using 6 strips and the trapezoidal rule. $235$ A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 0 & 5 & 10 & 15 & 20 & 25 \\ v & 14 & 12 & 18 & 14 & 9 & 11 \end{array}$ $\frac{655}{2}$ m Use the trapezoids the table makes to approximate $\int_{2}^{13} f (x) d x$ : $\begin{array}{cccccc} x & 2 & 5 & 8 & 11 & 13 \\ f (x) & 17 & 20 & 14 & 2 & 1 \end{array}$ $\frac{267}{2}$ Find $T_{2}$ for $\int_{0}^{4} x^{2} d x$ . Is it an overestimate or an underestimate? $\begin{array}{l} T_{2} = 24 \\ An overestimate \end{array}$ Bound the error of $T_{5}$ for $\int_{0}^{2} \cos 2 x d x$ . $∣ E_{T} ∣ \leq \frac{8}{75}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Trapezoidal RuleAnswer keyVersion 1

Date Period

Use the trapezoidal rule to answer each question.

Estimate the area under $y = 3 x^{2} - 3 x + 4$ from $x = 3$ to $x = 7$ using 4 strips and the trapezoidal rule. $274$
A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 2 & 4 & 6 & 8 & 10 & 12 \\ v & 11 & 17 & 17 & 17 & 13 & 12 \end{array}$ $151$ m
Water flows into a pond at the rates shown. Use trapezoids to estimate how much flows in from $t = 0$ to $t = 6$ : $\begin{array}{cccccc} t (h) & 0 & 1 & 3 & 4 & 6 \\ r (t) (gal/h) & 6 & 4 & 0 & 8 & 12 \end{array}$ $33$ gallons
Find $T_{2}$ for $\int_{0}^{2} \frac{6}{x + 1} d x$ . Is it an overestimate or an underestimate? $\begin{array}{l} T_{2} = 7 \\ An overestimate \end{array}$
Bound the error of $T_{8}$ for $\int_{0}^{1} x^{4} d x$ . $∣ E_{T} ∣ \leq \frac{1}{64}$
Estimate the area under $y = 3 x^{2} + x + 3$ from $x = 1$ to $x = 7$ using 6 strips and the trapezoidal rule. $387$
A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 5 & 10 & 15 & 20 & 25 & 30 \\ v & 19 & 15 & 10 & 15 & 13 & 18 \end{array}$ $\frac{715}{2}$ m
A car's velocity is read at the times shown. Use trapezoids to estimate how far it travels from $t = 0$ to $t = 9$ : $\begin{array}{cccccc} t (s) & 0 & 3 & 6 & 8 & 9 \\ v (t) (ft/s) & 11 & 7 & 20 & 13 & 10 \end{array}$ $112$ feet
Find $T_{2}$ for $\int_{0}^{2} (- x^{3} + 8) d x$ . Is it an overestimate or an underestimate? $\begin{array}{l} T_{2} = 11 \\ An underestimate \end{array}$
Bound the error of $T_{4}$ for $\int_{0}^{3} \sin x d x$ . $∣ E_{T} ∣ \leq \frac{9}{64}$
Estimate the area under $y = x^{2} - 3 x + 3$ from $x = 0$ to $x = 4$ using 4 strips and the trapezoidal rule. $10$
A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 0 & 1 & 2 & 3 & 4 & 5 \\ v & 14 & 16 & 11 & 10 & 7 & 11 \end{array}$ $\frac{113}{2}$ m
Use the trapezoids the table makes to approximate $\int_{1}^{7} f (x) d x$ : $\begin{array}{cccccc} x & 1 & 3 & 4 & 5 & 7 \\ f (x) & - 1 & 1 & 11 & 4 & - 3 \end{array}$ $\frac{29}{2}$
Find $T_{2}$ for $\int_{1}^{3} (x^{3} - 2 x^{2} + 5) d x$ . Is it an overestimate or an underestimate? $\begin{array}{l} T_{2} = 14 \\ An overestimate \end{array}$
Bound the error of $T_{6}$ for $\int_{1}^{3} \frac{1}{x} d x$ . $∣ E_{T} ∣ \leq \frac{1}{27}$
Estimate the area under $y = x^{2} + 2 x + 1$ from $x = 2$ to $x = 8$ using 6 strips and the trapezoidal rule. $235$
A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{ccccccc} t & 0 & 5 & 10 & 15 & 20 & 25 \\ v & 14 & 12 & 18 & 14 & 9 & 11 \end{array}$ $\frac{655}{2}$ m
Use the trapezoids the table makes to approximate $\int_{2}^{13} f (x) d x$ : $\begin{array}{cccccc} x & 2 & 5 & 8 & 11 & 13 \\ f (x) & 17 & 20 & 14 & 2 & 1 \end{array}$ $\frac{267}{2}$
Find $T_{2}$ for $\int_{0}^{4} x^{2} d x$ . Is it an overestimate or an underestimate? $\begin{array}{l} T_{2} = 24 \\ An overestimate \end{array}$
Bound the error of $T_{5}$ for $\int_{0}^{2} \cos 2 x d x$ . $∣ E_{T} ∣ \leq \frac{8}{75}$

How to do these

Find $T_{2}$ for $\int_{0}^{4} x^{2} d x$ .

Two trapezoids of width 2, with heights f(0) = 0, f(2) = 4 and f(4) = 16.
T_2 = (2/2)(0 + 2 · 4 + 16) = 24.
The exact value is 64/3, so 24 is an overestimate: x^2 is concave up.

The answer is $24$ .

Where students go wrong

Doubling the end heights instead of the inside ones. Every inside height is shared by two trapezoids, so it is the inside ones that count twice.

Also called trapezoid rule, trapezoidal approximation, trapezoidal sum or trapezoid sum.

What you can put on this worksheet

The trapezoidal rule: Estimate the area under $y = 3 x + 3$ from $x = 0$ to $x = 4$ using 4 strips and the trapezoidal rule. → $36$
The trapezoidal rule: A car's speed $v$ in m/s is recorded at these times $t$ in seconds. Estimate how far it travels, using the trapezoidal rule. $\begin{array}{cccccc} t & 0 & 2 & 4 & 6 & 8 \\ v & 16 & 11 & 11 & 12 & 9 \end{array}$ → $93$ m
A table with uneven widths: Use the trapezoids the table makes to approximate $\int_{1}^{6} f (x) d x$ : $\begin{array}{ccccc} x & 1 & 3 & 5 & 6 \\ f (x) & 11 & 5 & 20 & 7 \end{array}$ → $\frac{109}{2}$
Overestimate or underestimate: Does the trapezoidal rule overestimate or underestimate $\int_{1}^{4} \frac{1}{x} d x$ ? Why? → Overestimate: the curve is concave up
The error bound: $∣ f^{''} (x) ∣ \leq 9$ on $[1, 4]$ . Bound the error of $T_{2}$ for $\int_{1}^{4} f (x) d x$ . → $∣ E_{T} ∣ \leq \frac{81}{16}$

Questions about these worksheets

Yes. Every trapezoidal 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 trapezoidal 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 5: the trapezoidal rule, the trapezoidal rule, a table with uneven widths, overestimate or underestimate and the error bound. Tick as many as you want and set how many of each, or let it spread them evenly.

Doubling the end heights instead of the inside ones. Every inside height is shared by two trapezoids, so it is the inside ones that count twice.

Trapezoidal rule worksheet

Select your difficulty

The Trapezoidal Rule

The Trapezoidal RuleAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these trapezoidal rule worksheets free?+

Do the trapezoidal rule worksheets come with an answer key?+

Are these trapezoidal rule worksheets printable?+

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Can I get a different version of the same trapezoidal rule worksheet?+

Is this a free alternative to Kuta Software for trapezoidal rule?+

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What do students get wrong with trapezoidal rule?+