Accumulation and Average Value

What an integral means in context.

12 problems · about 21 minutes · CHA-4.D, CHA-4.E

NameDate

Figure — use these to answer the problems

The shaded area under a rate curve is the accumulated total over that interval.

y = -0.5x² + 3x + 0

a-0.5

Start time0

End time4

Figure 1

Straightforward practice. Get the method working first.

5 problems

The integral of f from 0 to 10 is 70. What is the average value?
Answer
2
What does the integral of a rate give?
- The accumulated total over the interval.
- The rate at the end of the interval.
Integral from 0 to 5 is 30. Average value?
Answer
Integral from 2 to 6 is 24. Average value?
Answer
A rate in litres per minute integrated over minutes. What unit results? 1 litres, 2 litres per minute.
Answer

The same ideas with more to keep track of.

3 problems

Velocity integrated over time gives what? 1 displacement, 2 acceleration.
Answer
For total distance rather than displacement, integrate what? 1 velocity, 2 speed.
Answer
A flow of 4 litres per minute for 20 minutes. Litres in total?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Match each integral with what it computes.
Draw a line from each item on the left to its match on the right.
- Integral of a flow rate over time
- Integral of velocity over time
- Integral divided by the interval width
- Total volume accumulated
- Displacement
- Average value
Integral from 0 to 8 is 40. Average value?
Answer
The Average
The integral of f from 0 to 4 is 20. What is the average value of f?
Answer
The Accumulation
A flow of 6 litres per minute for 15 minutes. How many litres in total?
AnswerL