Cogito
AP Calculus AB · Chapter 8 · Lesson 3
Accumulation and Average Value
What an integral means in context.
12 problems · about 21 minutes · CHA-4.D, CHA-4.E
What this lesson teaches
The student interprets definite integrals in context and computes average values.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThe integral of f from 0 to 10 is 70. What is the average value?
Answer 7
Why 7.
What does the integral of a rate give?
Answer The accumulated total over the interval.
Why The total accumulated.
Integral from 0 to 5 is 30. Average value?
Answer 6
Why 30 ÷ 5.
Integral from 2 to 6 is 24. Average value?
Answer 6
Why 24 ÷ 4.
A rate in litres per minute integrated over minutes. What unit results? 1 litres, 2 litres per minute.
Answer 1
Why The minutes cancel.
Build It Up
The same ideas with more to keep track of.
3 problemsVelocity integrated over time gives what? 1 displacement, 2 acceleration.
Answer 1
Why Rate times time.
For total distance rather than displacement, integrate what? 1 velocity, 2 speed.
Answer 2
Why Speed has no sign.
A flow of 4 litres per minute for 20 minutes. Litres in total?
Answer 80
Why 4 × 20.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each integral with what it computes.
Answer Integral of a flow rate over time → Total volume accumulated; Integral of velocity over time → Displacement; Integral divided by the interval width → Average value
Why Only one of them divides by anything.
Integral from 0 to 8 is 40. Average value?
Answer 5
Why 40 ÷ 8.
The Average: The integral of f from 0 to 4 is 20. What is the average value of f?
Answer 5
Why 5.
The Accumulation: A flow of 6 litres per minute for 15 minutes. How many litres in total?
Answer 90 L
Why 90 L.