The integral of a rate gives an accumulated total. Integrating litres per minute over minutes gives litres.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Units multiply
The unit of an integral is the integrand unit times the variable unit. That product names the quantity for you.
Net against total
An integral of velocity gives displacement, which can be zero after a round trip. Distance travelled integrates the speed.
Average value
The average value of f on [a, b] is the integral divided by (b − a).
Why divide
The integral is a total, and dividing by the width converts a total to a per-unit rate, which is what an average is.
Interpretation earns marks
The exam asks what an integral means in context. The answer is a sentence with units, not a number.
An integral of a rate is a total
Integrating a rate of flow gives volume; integrating a rate of sales gives revenue. Free-response questions supply a rate function and ask what the integral represents, with units.
Units come from integrand times variable
Litres per minute integrated with respect to minutes gives litres. Deriving the units this way is a reliable way to state what an integral means, and it is explicitly graded.
Average value
The average value over [a, b] is the integral divided by (b − a). It is the constant height giving the same accumulated total, which is exactly what an average ought to mean.
Net change against total amount
Integrating velocity gives displacement; integrating speed gives distance. The distinction between net and total recurs throughout applications and is a deliberate exam trap.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 0)
- Second point(4, 4)
- Slope between them1
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Marcus Interpret an Integral
Water flows into a tank at R(t) litres per minute, and Marcus integrates R from 0 to 10.
- Step 1
The integrand is measured in litres per minute.
Step 4: Your Turn
Practice makes it stick.
The Average
Problem 1 of 2
The integral of f from 0 to 4 is 20. What is the average value of f?
The Total
Problem 2 of 2
A flow of 6 litres per minute for 15 minutes. How many litres in total?
L
Read the Integral
1 of 8
Integral from 0 to 5 is 30. Average value?
2 of 8
Integral from 2 to 6 is 24. Average value?
3 of 8
A rate in litres per minute integrated over minutes. Which unit results? 1 litres, 2 litres per minute.
4 of 8
Velocity integrated over time gives what? 1 displacement, 2 acceleration.
5 of 8
For total distance rather than displacement, integrate what? 1 velocity, 2 speed.
6 of 8
A flow of 4 litres per minute for 20 minutes. Litres in total?
7 of 8
Match each integral with what it computes.
Tap a card on the left to start.
8 of 8
Integral from 0 to 8 is 40. Average value?
Step 5: Quick Check
Show what you know.
Question 1 of 2
The integral of f from 0 to 10 is 70. What is the average value?
Question 2 of 2
How do you name the quantity an integral computes?
What You Learned
- The integral of a rate is an accumulated total.
- The resulting unit is the integrand unit times the variable unit.
- Average value is the integral divided by the interval width.