Definite integrals worksheet

Find an antiderivative, put the top limit in, put the bottom limit in, subtract. That is the fundamental theorem, and it is why a question about area can be answered by algebra.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Definite Integrals Date Period Evaluate each integral, or rewrite it as asked. Evaluate the definite integral $\int_{1}^{3} ⁣ (4 - 27 x^{2}) d x$ $\int_{0}^{3} ∣ x^{2} - 1 ∣ d x$ $\int_{0}^{3} ⁣ (16 x - 18 x^{2}) d x$ $\int_{1}^{4} 5 \sqrt{x} d x$ $\int_{1}^{4} 9 x^{2} d x$ Evaluate by substitution, changing the bounds $\int_{0}^{1} - 12 x^{2} ⁣ {(2 x^{3} + 2)}^{2} d x$ $\int_{0}^{2} - 12 x ⁣ (3 x^{2} + 3) d x$ $\int_{0}^{3} 9 x^{2} ⁣ {(3 x^{3} + 9)}^{2} d x$ Integrate trig, exponential and $\frac{1}{x}$ functions $\int_{1}^{e} \frac{3}{x} d x$ $\int_{0}^{\frac{π}{2}} (3 \sin x - 5 \cos x) d x$ Evaluate integrals from a graph using areas The graph of $f$ is shown. Find $\int_{- 2}^{2} f (x) d x$ The graph of $f$ is shown. Find $\int_{3}^{4} f (x) d x$ Use properties of definite integrals Given $\int_{- 1}^{2} f (x) d x = - 5$ , $\int_{2}^{5} f (x) d x = 5$ and $\int_{- 1}^{2} g (x) d x = 6$ , find $\int_{- 1}^{5} f (x) d x$ Given $\int_{- 1}^{3} f (x) d x = 3$ , $\int_{3}^{4} f (x) d x = - 7$ and $\int_{- 1}^{3} g (x) d x = 3$ , find $\int_{- 1}^{4} f (x) d x$ Find how much the tank holds Water enters a tank at $R (t) = t + 5$ gallons per minute and leaves at $6$ gallons per minute. It holds $870$ gallons at $t = 0$ . How much does it hold at $t = 2$ ? Water flows into a tank at $r (t) = 5 t + 6$ gallons per minute. The tank holds $540$ gallons at $t = 0$ . How much does it hold at $t = 5$ ? Integrate absolute-value functions $\int_{- 3}^{0} ∣ x + 1 ∣ d x$ $\int_{- 4}^{5} ∣ x ∣ d x$ Rewrite in $u$ with new limits Use the substitution $u = x^{2} + 2$ to write $\int_{1}^{3} 2 x \sqrt{x^{2} + 2} d x$ as an integral in $u$ with new limits. Do not evaluate. Substitute $u = 3 + \sin x$ into $\int_{\frac{π}{6}}^{\frac{π}{2}} \cos x {(3 + \sin x)}^{3} d x$ and change the limits of integration. Do not evaluate. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Definite Integrals

Date Period

Evaluate each integral, or rewrite it as asked.

Evaluate the definite integral
$\int_{1}^{3} ⁣ (4 - 27 x^{2}) d x$
$\int_{0}^{3} ∣ x^{2} - 1 ∣ d x$
$\int_{0}^{3} ⁣ (16 x - 18 x^{2}) d x$
$\int_{1}^{4} 5 \sqrt{x} d x$
$\int_{1}^{4} 9 x^{2} d x$

Evaluate by substitution, changing the bounds
$\int_{0}^{1} - 12 x^{2} ⁣ {(2 x^{3} + 2)}^{2} d x$
$\int_{0}^{2} - 12 x ⁣ (3 x^{2} + 3) d x$
$\int_{0}^{3} 9 x^{2} ⁣ {(3 x^{3} + 9)}^{2} d x$

Integrate trig, exponential and $\frac{1}{x}$ functions
$\int_{1}^{e} \frac{3}{x} d x$
$\int_{0}^{\frac{π}{2}} (3 \sin x - 5 \cos x) d x$

Evaluate integrals from a graph using areas
The graph of $f$ is shown. Find $\int_{- 2}^{2} f (x) d x$
The graph of $f$ is shown. Find $\int_{3}^{4} f (x) d x$

Use properties of definite integrals
Given $\int_{- 1}^{2} f (x) d x = - 5$ , $\int_{2}^{5} f (x) d x = 5$ and $\int_{- 1}^{2} g (x) d x = 6$ , find $\int_{- 1}^{5} f (x) d x$
Given $\int_{- 1}^{3} f (x) d x = 3$ , $\int_{3}^{4} f (x) d x = - 7$ and $\int_{- 1}^{3} g (x) d x = 3$ , find $\int_{- 1}^{4} f (x) d x$

Find how much the tank holds
Water enters a tank at $R (t) = t + 5$ gallons per minute and leaves at $6$ gallons per minute. It holds $870$ gallons at $t = 0$ . How much does it hold at $t = 2$ ?
Water flows into a tank at $r (t) = 5 t + 6$ gallons per minute. The tank holds $540$ gallons at $t = 0$ . How much does it hold at $t = 5$ ?

Integrate absolute-value functions
$\int_{- 3}^{0} ∣ x + 1 ∣ d x$
$\int_{- 4}^{5} ∣ x ∣ d x$

Rewrite in $u$ with new limits
Use the substitution $u = x^{2} + 2$ to write $\int_{1}^{3} 2 x \sqrt{x^{2} + 2} d x$ as an integral in $u$ with new limits. Do not evaluate.
Substitute $u = 3 + \sin x$ into $\int_{\frac{π}{6}}^{\frac{π}{2}} \cos x {(3 + \sin x)}^{3} d x$ and change the limits of integration. Do not evaluate.



Sigma Prep · sigmaprep.io/worksheets Name Definite IntegralsAnswer keyVersion 1 Date Period Evaluate each integral, or rewrite it as asked. Evaluate the definite integral $\int_{1}^{3} ⁣ (4 - 27 x^{2}) d x$ $- 226$ $\int_{0}^{3} ∣ x^{2} - 1 ∣ d x$ $\frac{22}{3}$ $\int_{0}^{3} ⁣ (16 x - 18 x^{2}) d x$ $- 90$ $\int_{1}^{4} 5 \sqrt{x} d x$ $\frac{70}{3}$ $\int_{1}^{4} 9 x^{2} d x$ $189$ Evaluate by substitution, changing the bounds $\int_{0}^{1} - 12 x^{2} ⁣ {(2 x^{3} + 2)}^{2} d x$ $- \frac{112}{3}$ $\int_{0}^{2} - 12 x ⁣ (3 x^{2} + 3) d x$ $- 216$ $\int_{0}^{3} 9 x^{2} ⁣ {(3 x^{3} + 9)}^{2} d x$ $242757$ Integrate trig, exponential and $\frac{1}{x}$ functions $\int_{1}^{e} \frac{3}{x} d x$ $3$ $\int_{0}^{\frac{π}{2}} (3 \sin x - 5 \cos x) d x$ $- 2$ Evaluate integrals from a graph using areas The graph of $f$ is shown. Find $\int_{- 2}^{2} f (x) d x$ $- 1$ The graph of $f$ is shown. Find $\int_{3}^{4} f (x) d x$ $- \frac{1}{2}$ Use properties of definite integrals Given $\int_{- 1}^{2} f (x) d x = - 5$ , $\int_{2}^{5} f (x) d x = 5$ and $\int_{- 1}^{2} g (x) d x = 6$ , find $\int_{- 1}^{5} f (x) d x$ $0$ Given $\int_{- 1}^{3} f (x) d x = 3$ , $\int_{3}^{4} f (x) d x = - 7$ and $\int_{- 1}^{3} g (x) d x = 3$ , find $\int_{- 1}^{4} f (x) d x$ $- 4$ Find how much the tank holds Water enters a tank at $R (t) = t + 5$ gallons per minute and leaves at $6$ gallons per minute. It holds $870$ gallons at $t = 0$ . How much does it hold at $t = 2$ ? $870$ gallons Water flows into a tank at $r (t) = 5 t + 6$ gallons per minute. The tank holds $540$ gallons at $t = 0$ . How much does it hold at $t = 5$ ? $632.5$ gallons Integrate absolute-value functions $\int_{- 3}^{0} ∣ x + 1 ∣ d x$ $\frac{5}{2}$ $\int_{- 4}^{5} ∣ x ∣ d x$ $\frac{41}{2}$ Rewrite in $u$ with new limits Use the substitution $u = x^{2} + 2$ to write $\int_{1}^{3} 2 x \sqrt{x^{2} + 2} d x$ as an integral in $u$ with new limits. Do not evaluate. $\int_{3}^{11} \sqrt{u} d u$ Substitute $u = 3 + \sin x$ into $\int_{\frac{π}{6}}^{\frac{π}{2}} \cos x {(3 + \sin x)}^{3} d x$ and change the limits of integration. Do not evaluate. $\int_{\frac{7}{2}}^{4} u^{3} d u$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Definite IntegralsAnswer keyVersion 1

Date Period

Evaluate each integral, or rewrite it as asked.

Evaluate the definite integral
$\int_{1}^{3} ⁣ (4 - 27 x^{2}) d x$ $- 226$
$\int_{0}^{3} ∣ x^{2} - 1 ∣ d x$ $\frac{22}{3}$
$\int_{0}^{3} ⁣ (16 x - 18 x^{2}) d x$ $- 90$
$\int_{1}^{4} 5 \sqrt{x} d x$ $\frac{70}{3}$
$\int_{1}^{4} 9 x^{2} d x$ $189$

Evaluate by substitution, changing the bounds
$\int_{0}^{1} - 12 x^{2} ⁣ {(2 x^{3} + 2)}^{2} d x$ $- \frac{112}{3}$
$\int_{0}^{2} - 12 x ⁣ (3 x^{2} + 3) d x$ $- 216$
$\int_{0}^{3} 9 x^{2} ⁣ {(3 x^{3} + 9)}^{2} d x$ $242757$

Integrate trig, exponential and $\frac{1}{x}$ functions
$\int_{1}^{e} \frac{3}{x} d x$ $3$
$\int_{0}^{\frac{π}{2}} (3 \sin x - 5 \cos x) d x$ $- 2$

Evaluate integrals from a graph using areas
The graph of $f$ is shown. Find $\int_{- 2}^{2} f (x) d x$ $- 1$
The graph of $f$ is shown. Find $\int_{3}^{4} f (x) d x$ $- \frac{1}{2}$

Use properties of definite integrals
Given $\int_{- 1}^{2} f (x) d x = - 5$ , $\int_{2}^{5} f (x) d x = 5$ and $\int_{- 1}^{2} g (x) d x = 6$ , find $\int_{- 1}^{5} f (x) d x$ $0$
Given $\int_{- 1}^{3} f (x) d x = 3$ , $\int_{3}^{4} f (x) d x = - 7$ and $\int_{- 1}^{3} g (x) d x = 3$ , find $\int_{- 1}^{4} f (x) d x$ $- 4$

Find how much the tank holds
Water enters a tank at $R (t) = t + 5$ gallons per minute and leaves at $6$ gallons per minute. It holds $870$ gallons at $t = 0$ . How much does it hold at $t = 2$ ? $870$ gallons
Water flows into a tank at $r (t) = 5 t + 6$ gallons per minute. The tank holds $540$ gallons at $t = 0$ . How much does it hold at $t = 5$ ? $632.5$ gallons

Integrate absolute-value functions
$\int_{- 3}^{0} ∣ x + 1 ∣ d x$ $\frac{5}{2}$
$\int_{- 4}^{5} ∣ x ∣ d x$ $\frac{41}{2}$

Rewrite in $u$ with new limits
Use the substitution $u = x^{2} + 2$ to write $\int_{1}^{3} 2 x \sqrt{x^{2} + 2} d x$ as an integral in $u$ with new limits. Do not evaluate. $\int_{3}^{11} \sqrt{u} d u$
Substitute $u = 3 + \sin x$ into $\int_{\frac{π}{6}}^{\frac{π}{2}} \cos x {(3 + \sin x)}^{3} d x$ and change the limits of integration. Do not evaluate. $\int_{\frac{7}{2}}^{4} u^{3} d u$

How to do these

$\int_{0}^{2} 3 x^{2} d x$

The antiderivative is x cubed.
At 2 that is 8, at 0 it is 0.
Subtract.

The answer is $8$ .

Where students go wrong

Subtracting the wrong way round. It is top minus bottom, and swapping them flips the sign of every answer on the page.

Also called fundamental theorem of calculus, area under a curve or evaluating integrals.

What you can put on this worksheet

Definite integrals: $\int_{1}^{3} (4 x + 2) d x$ → $20$
The area under a curve: Area under $y = - 2 x^{2} + 2 x + 4$ from $x = - 1$ to $x = 2$ → $9$
Substitution with changed bounds: $\int_{0}^{2} - 24 x ⁣ (3 x^{2} + 1) d x$ → $- 336$
Trig, exponential and 1/x: $\int_{1}^{e} \frac{4}{x} d x$ → $4$
From a graph, by areas: The graph of $f$ is shown. Find $\int_{2}^{7} f (x) d x$ — the graph of f on [-1, 8] → $\frac{1}{2}$
Properties, from given values: Given $\int_{- 1}^{2} f (x) d x = - 5$ , $\int_{2}^{6} f (x) d x = 3$ and $\int_{- 1}^{2} g (x) d x = 4$ , find $\int_{- 1}^{6} f (x) d x$ → $- 2$
Net change: rates in and out: Water flows into a tank at $r (t) = t + 7$ gallons per minute. The tank holds $550$ gallons at $t = 0$ . How much does it hold at $t = 4$ ? → $586$ gallons
Absolute-value integrands: $\int_{- 2}^{5} ∣ x - 1 ∣ d x$ → $\frac{25}{2}$
Roots, logs on negative bounds, |x^2 - c^2|, pieces: $\int_{- 8}^{- 2} \frac{4}{x} d x$ → $- 4 \ln 4$
Rewrite in u with new limits: Rewrite $\int_{1}^{3} \frac{2 x}{{(x^{2} + 3)}^{4}} d x$ in terms of $u = x^{2} + 3$ . Do not evaluate. → $\int_{4}^{12} \frac{1}{u^{4}} d u$

Questions about these worksheets

Yes. Every definite integrals 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 definite integrals 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 10: definite integrals, the area under a curve, substitution with changed bounds, trig, exponential and 1/x, from a graph, by areas, properties, from given values, net change: rates in and out, absolute-value integrands, roots, logs on negative bounds, |x^2 - c^2|, pieces and rewrite in u with new limits. Tick as many as you want and set how many of each, or let it spread them evenly.

Subtracting the wrong way round. It is top minus bottom, and swapping them flips the sign of every answer on the page.

Definite integrals worksheet

Select your difficulty

Evaluate the definite integral

Evaluate by substitution, changing the bounds

Integrate trig, exponential and 1x\frac{1}{x}x1​ functions

Evaluate integrals from a graph using areas

Use properties of definite integrals

Find how much the tank holds

Integrate absolute-value functions

Rewrite in uuu with new limits

Evaluate the definite integral

Evaluate by substitution, changing the bounds

Integrate trig, exponential and 1x\frac{1}{x}x1​ functions

Evaluate integrals from a graph using areas

Use properties of definite integrals

Find how much the tank holds

Integrate absolute-value functions

Rewrite in uuu with new limits

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these definite integrals worksheets free?+

Do the definite integrals worksheets come with an answer key?+

Are these definite integrals worksheets printable?+

What grade are these definite integrals worksheets for?+

Can I get a different version of the same definite integrals worksheet?+

Is this a free alternative to Kuta Software for definite integrals?+

Can I choose how hard the definite integrals questions are?+

What kinds of question are on a definite integrals worksheet?+

What do students get wrong with definite integrals?+

Integrate trig, exponential and $\frac{1}{x}$ functions

Rewrite in $u$ with new limits

Integrate trig, exponential and $\frac{1}{x}$ functions

Rewrite in $u$ with new limits