Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Build a total from an initial amount and accumulated change.
- Evaluate a polynomial definite integral.
- Check the result against simple rate bounds.
Before you start
Antiderivatives of constants and powers, and evaluating expressions at endpoints.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A tank starts with 5 litres. Water enters at r(t) = 2 + 3t litres per minute for 0 ≤ t ≤ 4. How much water is in the tank at four minutes?
Why this math matters
Rate measurements tell us how quickly something changes, not how much is already present. Multiplying by time works for a constant rate, but a changing rate must be accumulated across the interval. A definite integral performs that accumulation and tracks the units automatically: litres per minute times minutes becomes litres.
Set up the model
A useful answer starts with clear assumptions:
- The tank has enough capacity and no leaks or outflow.
- The inflow follows the given formula throughout the four minutes.
- Time is in minutes and the initial volume is exactly five litres.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
How does a changing flow rate fill a tank?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A tank starts with 5 litres. Water enters at r(t) = 2 + 3t litres per minute for 0 ≤ t ≤ 4. How much water is in the tank at four minutes?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Write initial amount plus change
V(4) = 5 + ∫₀⁴(2 + 3t) dt
The integral counts only water entering during the interval. The five litres already present must be added separately.
Find and evaluate an antiderivative
∫₀⁴(2 + 3t) dt = [2t + 1.5t²]₀⁴ = 8 + 24 = 32 L
The power rule for integration reverses differentiation. Evaluating the upper value minus the lower value gives the accumulated inflow.
Add and check
V(4) = 5 + 32 = 37 L; average inflow = 32/4 = 8 L/min
The rate rises linearly from two to fourteen litres per minute. Their average is eight, independently confirming the integral for this linear model.
The result
The tank contains 37 litres after four minutes: 5 initially present plus 32 added.
With simultaneous outflow, integrate inflow minus outflow to obtain net change. A negative net rate removes water; the physical volume must still stay nonnegative on the modeled interval.
Common mistakes to catch
- Multiplying the final rate, 14, by all four minutes overcounts the earlier slower inflow.
- Reporting 32 litres as the final volume forgets the initial five litres.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the original tank's volume at t = 2?
Show a hint
Evaluate 5 + 2t + 1.5t² at two.
Reveal answer and explanation
15 litres
5 + 4 + 6 = 15 litres; ten litres have entered.
Practice 2
A separate tank gains water at 3 L/min for two minutes, then loses it at 1 L/min for two minutes. What is its net change?
Show a hint
Keep the second rate negative.
Reveal answer and explanation
+4 litres
The signed accumulation is 3(2) − 1(2) = 4 litres, assuming enough water remains for the outflow.
Take the idea with you
Use the same initial-plus-integral structure for position from velocity, charge from current, or inventory from net delivery rate.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Can three readings estimate the total flow?
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