The balance of an account changes at r(t) = -t^2 + 3t - 2 dollars per day, positive for deposits and negative for withdrawals. For 0 ≤ t ≤ 3, find the net change and the total change.

Net change: -3/2 dollars; Total change: 11/6 dollars

The depth of a pond changes at r(t) = 3t^2 - 15t + 12 inches per week, positive while it rises and negative while it falls. For 0 ≤ t ≤ 6, find the net change and the total change.

Net change: 18 inches; Total change: 45 inches

The volume of water in a tank changes at r(t) = t^2 - 7t + 12 gallons per minute, positive while it fills and negative while it drains. For 0 ≤ t ≤ 5, find the net change and the total change.

Net change: 85/6 gallons; Total change: 29/2 gallons

The volume of water in a tank changes at r(t) = 2t^2 - 14t + 24 gallons per minute, positive while it fills and negative while it drains. For 0 ≤ t ≤ 5, find the net change and the total change.

Net change: 85/3 gallons; Total change: 29 gallons

The balance of an account changes at r(t) = 3t^2 - 9t + 6 dollars per day, positive for deposits and negative for withdrawals. For 0 ≤ t ≤ 3, find the net change and the total change.

Net change: 9/2 dollars; Total change: 11/2 dollars

The balance of an account changes at r(t) = t^2 - 6t + 8 dollars per day, positive for deposits and negative for withdrawals. For 0 ≤ t ≤ 5, find the net change and the total change.

Net change: 20/3 dollars; Total change: 28/3 dollars

The balance of an account changes at r(t) = 3t - 3 dollars per day, positive for deposits and negative for withdrawals. For 0 ≤ t ≤ 3, find the net change and the total change.

Net change: 9/2 dollars; Total change: 15/2 dollars

Worksheets · Calculus

Net change theorem worksheet

The integral of a rate is the change in the amount. Add the starting amount and you have the amount at the end; integrate the absolute value and you have everything that moved.

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.



Sigma Prep · sigmaprep.io/worksheets Name Net Change Date Period Answer each question. Include units. Water enters a tank at $R (t) = 6 t + 6$ gallons per minute and leaves at $12$ gallons per minute. It holds $100$ gallons at $t = 0$ . How much does it hold at $t = 6$ ? The marginal cost of making $x$ units is $C^{'} (x) = 50 - \frac{8 x}{50}$ dollars per unit. How much does raising production from $30$ to $50$ units cost? The balance of an account changes at $r (t) = - t^{2} + 3 t - 2$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change. Water enters a tank at $R (t) = t + 2$ gallons per minute and leaves at $2$ gallons per minute. It holds $360$ gallons at $t = 0$ . How much does it hold at $t = 5$ ? A town grows at a rate of $r (t) = 3 t^{2} + 6$ people per year. How many people are added from $t = 2$ to $t = 3$ ? The depth of a pond changes at $r (t) = 3 t^{2} - 15 t + 12$ inches per week, positive while it rises and negative while it falls. For $0 \leq t \leq 6$ , find the net change and the total change. Water enters a tank at $R (t) = t + 3$ gallons per minute and leaves at $5$ gallons per minute. It holds $120$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? A reservoir fills at a rate of $r (t) = 9 t^{2} - 6 t + 6$ acre-feet per day. The reservoir holds $360$ acre-feet at $t = 0$ . How many acre-feet at $t = 3$ ? The volume of water in a tank changes at $r (t) = t^{2} - 7 t + 12$ gallons per minute, positive while it fills and negative while it drains. For $0 \leq t \leq 5$ , find the net change and the total change. Water enters a tank at $R (t) = 5 t + 6$ gallons per minute and leaves at $12$ gallons per minute. It holds $720$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? Sand is poured onto a pile at a rate of $r (t) = 6 t^{2} - 6 t + 3$ cubic feet per hour. The pile holds $130$ cubic feet at $t = 0$ . How many cubic feet at $t = 2$ ? The volume of water in a tank changes at $r (t) = 2 t^{2} - 14 t + 24$ gallons per minute, positive while it fills and negative while it drains. For $0 \leq t \leq 5$ , find the net change and the total change. Water enters a tank at $R (t) = 3 t + 5$ gallons per minute and leaves at $10$ gallons per minute. It holds $720$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? A reservoir fills at a rate of $r (t) = 6 t^{2} + 8$ acre-feet per day. How many acre-feet are added from $t = 1$ to $t = 2$ ? The balance of an account changes at $r (t) = 3 t^{2} - 9 t + 6$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change. Water enters a tank at $R (t) = t + 7$ gallons per minute and leaves at $8$ gallons per minute. It holds $570$ gallons at $t = 0$ . How much does it hold at $t = 3$ ? Water flows into a tank at a rate of $r (t) = 3 t^{2} + 2$ gallons per minute. How many gallons are added from $t = 1$ to $t = 3$ ? The balance of an account changes at $r (t) = t^{2} - 6 t + 8$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 5$ , find the net change and the total change. Water enters a tank at $R (t) = 3 t + 4$ gallons per minute and leaves at $3$ gallons per minute. It holds $310$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? A town grows at a rate of $r (t) = 9 t^{2} + 6$ people per year. The town has $340$ people at $t = 0$ . How many people at $t = 5$ ? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Net Change

Date Period

Answer each question. Include units.

Water enters a tank at $R (t) = 6 t + 6$ gallons per minute and leaves at $12$ gallons per minute. It holds $100$ gallons at $t = 0$ . How much does it hold at $t = 6$ ?
The marginal cost of making $x$ units is $C^{'} (x) = 50 - \frac{8 x}{50}$ dollars per unit. How much does raising production from $30$ to $50$ units cost?
The balance of an account changes at $r (t) = - t^{2} + 3 t - 2$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change.
Water enters a tank at $R (t) = t + 2$ gallons per minute and leaves at $2$ gallons per minute. It holds $360$ gallons at $t = 0$ . How much does it hold at $t = 5$ ?
A town grows at a rate of $r (t) = 3 t^{2} + 6$ people per year. How many people are added from $t = 2$ to $t = 3$ ?
The depth of a pond changes at $r (t) = 3 t^{2} - 15 t + 12$ inches per week, positive while it rises and negative while it falls. For $0 \leq t \leq 6$ , find the net change and the total change.
Water enters a tank at $R (t) = t + 3$ gallons per minute and leaves at $5$ gallons per minute. It holds $120$ gallons at $t = 0$ . How much does it hold at $t = 7$ ?
A reservoir fills at a rate of $r (t) = 9 t^{2} - 6 t + 6$ acre-feet per day. The reservoir holds $360$ acre-feet at $t = 0$ . How many acre-feet at $t = 3$ ?
The volume of water in a tank changes at $r (t) = t^{2} - 7 t + 12$ gallons per minute, positive while it fills and negative while it drains. For $0 \leq t \leq 5$ , find the net change and the total change.
Water enters a tank at $R (t) = 5 t + 6$ gallons per minute and leaves at $12$ gallons per minute. It holds $720$ gallons at $t = 0$ . How much does it hold at $t = 7$ ?
Sand is poured onto a pile at a rate of $r (t) = 6 t^{2} - 6 t + 3$ cubic feet per hour. The pile holds $130$ cubic feet at $t = 0$ . How many cubic feet at $t = 2$ ?
The volume of water in a tank changes at $r (t) = 2 t^{2} - 14 t + 24$ gallons per minute, positive while it fills and negative while it drains. For $0 \leq t \leq 5$ , find the net change and the total change.
Water enters a tank at $R (t) = 3 t + 5$ gallons per minute and leaves at $10$ gallons per minute. It holds $720$ gallons at $t = 0$ . How much does it hold at $t = 7$ ?
A reservoir fills at a rate of $r (t) = 6 t^{2} + 8$ acre-feet per day. How many acre-feet are added from $t = 1$ to $t = 2$ ?
The balance of an account changes at $r (t) = 3 t^{2} - 9 t + 6$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change.
Water enters a tank at $R (t) = t + 7$ gallons per minute and leaves at $8$ gallons per minute. It holds $570$ gallons at $t = 0$ . How much does it hold at $t = 3$ ?
Water flows into a tank at a rate of $r (t) = 3 t^{2} + 2$ gallons per minute. How many gallons are added from $t = 1$ to $t = 3$ ?
The balance of an account changes at $r (t) = t^{2} - 6 t + 8$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 5$ , find the net change and the total change.
Water enters a tank at $R (t) = 3 t + 4$ gallons per minute and leaves at $3$ gallons per minute. It holds $310$ gallons at $t = 0$ . How much does it hold at $t = 7$ ?
A town grows at a rate of $r (t) = 9 t^{2} + 6$ people per year. The town has $340$ people at $t = 0$ . How many people at $t = 5$ ?



Sigma Prep · sigmaprep.io/worksheets Name Net ChangeAnswer keyVersion 1 Date Period Answer each question. Include units. Water enters a tank at $R (t) = 6 t + 6$ gallons per minute and leaves at $12$ gallons per minute. It holds $100$ gallons at $t = 0$ . How much does it hold at $t = 6$ ? $172$ gallons The marginal cost of making $x$ units is $C^{'} (x) = 50 - \frac{8 x}{50}$ dollars per unit. How much does raising production from $30$ to $50$ units cost? $$ 872$ The balance of an account changes at $r (t) = - t^{2} + 3 t - 2$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change. $\begin{array}{l} Net change: - \frac{3}{2} dollars \\ Total change: \frac{11}{6} dollars \end{array}$ Water enters a tank at $R (t) = t + 2$ gallons per minute and leaves at $2$ gallons per minute. It holds $360$ gallons at $t = 0$ . How much does it hold at $t = 5$ ? $\frac{745}{2}$ gallons A town grows at a rate of $r (t) = 3 t^{2} + 6$ people per year. How many people are added from $t = 2$ to $t = 3$ ? $25$ people The depth of a pond changes at $r (t) = 3 t^{2} - 15 t + 12$ inches per week, positive while it rises and negative while it falls. For $0 \leq t \leq 6$ , find the net change and the total change. $\begin{array}{l} Net change: 18 inches \\ Total change: 45 inches \end{array}$ Water enters a tank at $R (t) = t + 3$ gallons per minute and leaves at $5$ gallons per minute. It holds $120$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? $\frac{261}{2}$ gallons A reservoir fills at a rate of $r (t) = 9 t^{2} - 6 t + 6$ acre-feet per day. The reservoir holds $360$ acre-feet at $t = 0$ . How many acre-feet at $t = 3$ ? $432$ acre-feet The volume of water in a tank changes at $r (t) = t^{2} - 7 t + 12$ gallons per minute, positive while it fills and negative while it drains. For $0 \leq t \leq 5$ , find the net change and the total change. $\begin{array}{l} Net change: \frac{85}{6} gallons \\ Total change: \frac{29}{2} gallons \end{array}$ Water enters a tank at $R (t) = 5 t + 6$ gallons per minute and leaves at $12$ gallons per minute. It holds $720$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? $\frac{1601}{2}$ gallons Sand is poured onto a pile at a rate of $r (t) = 6 t^{2} - 6 t + 3$ cubic feet per hour. The pile holds $130$ cubic feet at $t = 0$ . How many cubic feet at $t = 2$ ? $140$ cubic feet The volume of water in a tank changes at $r (t) = 2 t^{2} - 14 t + 24$ gallons per minute, positive while it fills and negative while it drains. For $0 \leq t \leq 5$ , find the net change and the total change. $\begin{array}{l} Net change: \frac{85}{3} gallons \\ Total change: 29 gallons \end{array}$ Water enters a tank at $R (t) = 3 t + 5$ gallons per minute and leaves at $10$ gallons per minute. It holds $720$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? $\frac{1517}{2}$ gallons A reservoir fills at a rate of $r (t) = 6 t^{2} + 8$ acre-feet per day. How many acre-feet are added from $t = 1$ to $t = 2$ ? $22$ acre-feet The balance of an account changes at $r (t) = 3 t^{2} - 9 t + 6$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change. $\begin{array}{l} Net change: \frac{9}{2} dollars \\ Total change: \frac{11}{2} dollars \end{array}$ Water enters a tank at $R (t) = t + 7$ gallons per minute and leaves at $8$ gallons per minute. It holds $570$ gallons at $t = 0$ . How much does it hold at $t = 3$ ? $\frac{1143}{2}$ gallons Water flows into a tank at a rate of $r (t) = 3 t^{2} + 2$ gallons per minute. How many gallons are added from $t = 1$ to $t = 3$ ? $30$ gallons The balance of an account changes at $r (t) = t^{2} - 6 t + 8$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 5$ , find the net change and the total change. $\begin{array}{l} Net change: \frac{20}{3} dollars \\ Total change: \frac{28}{3} dollars \end{array}$ Water enters a tank at $R (t) = 3 t + 4$ gallons per minute and leaves at $3$ gallons per minute. It holds $310$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? $\frac{781}{2}$ gallons A town grows at a rate of $r (t) = 9 t^{2} + 6$ people per year. The town has $340$ people at $t = 0$ . How many people at $t = 5$ ? $745$ people Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Net ChangeAnswer keyVersion 1

Date Period

Answer each question. Include units.

Water enters a tank at $R (t) = 6 t + 6$ gallons per minute and leaves at $12$ gallons per minute. It holds $100$ gallons at $t = 0$ . How much does it hold at $t = 6$ ? $172$ gallons
The marginal cost of making $x$ units is $C^{'} (x) = 50 - \frac{8 x}{50}$ dollars per unit. How much does raising production from $30$ to $50$ units cost? $$ 872$
The balance of an account changes at $r (t) = - t^{2} + 3 t - 2$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change. $\begin{array}{l} Net change: - \frac{3}{2} dollars \\ Total change: \frac{11}{6} dollars \end{array}$
Water enters a tank at $R (t) = t + 2$ gallons per minute and leaves at $2$ gallons per minute. It holds $360$ gallons at $t = 0$ . How much does it hold at $t = 5$ ? $\frac{745}{2}$ gallons
A town grows at a rate of $r (t) = 3 t^{2} + 6$ people per year. How many people are added from $t = 2$ to $t = 3$ ? $25$ people
The depth of a pond changes at $r (t) = 3 t^{2} - 15 t + 12$ inches per week, positive while it rises and negative while it falls. For $0 \leq t \leq 6$ , find the net change and the total change. $\begin{array}{l} Net change: 18 inches \\ Total change: 45 inches \end{array}$
Water enters a tank at $R (t) = t + 3$ gallons per minute and leaves at $5$ gallons per minute. It holds $120$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? $\frac{261}{2}$ gallons
A reservoir fills at a rate of $r (t) = 9 t^{2} - 6 t + 6$ acre-feet per day. The reservoir holds $360$ acre-feet at $t = 0$ . How many acre-feet at $t = 3$ ? $432$ acre-feet
The volume of water in a tank changes at $r (t) = t^{2} - 7 t + 12$ gallons per minute, positive while it fills and negative while it drains. For $0 \leq t \leq 5$ , find the net change and the total change. $\begin{array}{l} Net change: \frac{85}{6} gallons \\ Total change: \frac{29}{2} gallons \end{array}$
Water enters a tank at $R (t) = 5 t + 6$ gallons per minute and leaves at $12$ gallons per minute. It holds $720$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? $\frac{1601}{2}$ gallons
Sand is poured onto a pile at a rate of $r (t) = 6 t^{2} - 6 t + 3$ cubic feet per hour. The pile holds $130$ cubic feet at $t = 0$ . How many cubic feet at $t = 2$ ? $140$ cubic feet
The volume of water in a tank changes at $r (t) = 2 t^{2} - 14 t + 24$ gallons per minute, positive while it fills and negative while it drains. For $0 \leq t \leq 5$ , find the net change and the total change. $\begin{array}{l} Net change: \frac{85}{3} gallons \\ Total change: 29 gallons \end{array}$
Water enters a tank at $R (t) = 3 t + 5$ gallons per minute and leaves at $10$ gallons per minute. It holds $720$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? $\frac{1517}{2}$ gallons
A reservoir fills at a rate of $r (t) = 6 t^{2} + 8$ acre-feet per day. How many acre-feet are added from $t = 1$ to $t = 2$ ? $22$ acre-feet
The balance of an account changes at $r (t) = 3 t^{2} - 9 t + 6$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change. $\begin{array}{l} Net change: \frac{9}{2} dollars \\ Total change: \frac{11}{2} dollars \end{array}$
Water enters a tank at $R (t) = t + 7$ gallons per minute and leaves at $8$ gallons per minute. It holds $570$ gallons at $t = 0$ . How much does it hold at $t = 3$ ? $\frac{1143}{2}$ gallons
Water flows into a tank at a rate of $r (t) = 3 t^{2} + 2$ gallons per minute. How many gallons are added from $t = 1$ to $t = 3$ ? $30$ gallons
The balance of an account changes at $r (t) = t^{2} - 6 t + 8$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 5$ , find the net change and the total change. $\begin{array}{l} Net change: \frac{20}{3} dollars \\ Total change: \frac{28}{3} dollars \end{array}$
Water enters a tank at $R (t) = 3 t + 4$ gallons per minute and leaves at $3$ gallons per minute. It holds $310$ gallons at $t = 0$ . How much does it hold at $t = 7$ ? $\frac{781}{2}$ gallons
A town grows at a rate of $r (t) = 9 t^{2} + 6$ people per year. The town has $340$ people at $t = 0$ . How many people at $t = 5$ ? $745$ people

How to do these

Water flows into a tank at $r (t) = 2 t + 5$ gallons per minute. The tank holds $40$ gallons at $t = 0$ . How many at $t = 4$ ?

The water added is the integral of 2t + 5 from 0 to 4, which is 16 + 20 = 36 gallons.
Add what was there to start: 40 + 36 = 76.

The answer is $76$ gallons.

Where students go wrong

Reporting the integral of the rate as the amount. It is only the change; the starting amount still has to be added.

Also called net change theorem, accumulation from a rate, rate in and rate out problems or integrating rates of change.

What you can put on this worksheet

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
Amount from a rate, in context: Sand is poured onto a pile at a rate of $r (t) = 2 t + 10$ cubic feet per hour. The pile holds $230$ cubic feet at $t = 0$ . How many cubic feet at $t = 4$ ? → $286$ cubic feet
Net change and total change: The balance of an account changes at $r (t) = 3 t - 3$ dollars per day, positive for deposits and negative for withdrawals. For $0 \leq t \leq 3$ , find the net change and the total change. → $\begin{array}{l} Net change: \frac{9}{2} dollars \\ Total change: \frac{15}{2} dollars \end{array}$

Questions about these worksheets

Yes. Every net change theorem 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 net change theorem 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: net change: rates in and out, amount from a rate, in context and net change and total change. Tick as many as you want and set how many of each, or let it spread them evenly.

Reporting the integral of the rate as the amount. It is only the change; the starting amount still has to be added.

Net change theorem worksheet

Select your difficulty

Net Change

Net ChangeAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these net change theorem worksheets free?+

Do the net change theorem worksheets come with an answer key?+

Are these net change theorem worksheets printable?+

What grade are these net change theorem worksheets for?+

Can I get a different version of the same net change theorem worksheet?+

Is this a free alternative to Kuta Software for net change theorem?+

Can I choose how hard the net change theorem questions are?+

What kinds of question are on a net change theorem worksheet?+

What do students get wrong with net change theorem?+