The integral of a rate gives the accumulated total. Integrating a flow rate in litres per minute 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. Litres per minute times minutes gives litres.
Net against total
An integral of velocity gives displacement, which can be zero after a round trip. For distance travelled, integrate 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 it to a per-unit rate, which is what an average is.
Interpretation earns marks
Free-response questions ask what an integral means in context. The answer needs a sentence with units, not a number.
What an integral means in context
The integral of a rate gives an accumulated total, with units that follow from the integrand times the variable. Exam answers must state that meaning in a sentence, not just report a number.
Accumulation functions
Defining F(x) as the integral of f from a to x builds a new function whose derivative is f. Questions about where F increases, has extrema or is concave up are answered by reading the graph of f.
Average value of a function
The average value over [a, b] is the integral divided by (b − a). It is the constant height a rectangle would need to have the same area, which is exactly what an average should mean.
Average value against average rate
The average value of a function and the average rate of change of a function are different quantities. The first integrates f; the second differences f. Reading which is asked for is essential.
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 Diego Interpret an Integral
Water flows into a tank at R(t) litres per minute, and Diego computes the integral of 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 Accumulation
Problem 2 of 2
A flow of 6 litres per minute for 15 minutes. How many litres in total?
L
Interpret and Average
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. What 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
What does the integral of a rate give?
What You Learned
- The integral of a rate gives an accumulated total.
- The unit is the integrand unit times the variable unit.
- Average value is the integral divided by the interval width.