Integration is an essential tool in the physical sciences, in economics and in statistics. It is needed when continuously varying quantities are involved in a mathematical model of a real-world process. It has wide ranging applications such as finding complex areas, volumes, probabilities, centre of mass, kinematics, average value of a function and net change for a quantity given the rate of change.
Net change
Given a function involving the rate of change of a quantity, integration allows us to calculate the net change of the quantity over a given interval or find an explicit formula for the quantity, given initial conditions. For example, if a function described the rate of water flowing in and out of a tank over time, the integral of such a function would allow us to find the net change in volume of water in the tank for a given interval.
Another example would be the integral of a velocity function (the rate of change of displacement with respect to time) would give us the net change in displacement over a given time interval.
Worked examples
Example 1
A population, $$P(t), of fish in a pond is known to vary according to the function $$P′(t)=300sin(πt6), where $$t is measured in months since counting began.
(a) What is the net change in population in the first $$4 months since counting began?
Think: We have been given the rate of change of the population, so to find the net change we need to find the value of the definite integral from $$t=0 to $$t=4.
Do:
| Net change | $$= | $$∫40300sin(πt6)dt |
| $$= | $$[−1800πcos(πt6)]40 | |
| $$= | $$−1800πcos2π3+1800πcos0 | |
| $$≈ | $$859 |
Thus, the population increased by approximately $$859 fish in the first $$4 months.
(b) What is the average rate of change of the population between $$t=3 and $$t=10?
Think: The average rate of change will be the net change in population over the interval divided by the change in time.
Do:
| Average rate of change | $$= | $$∫103300sin(πt6)dt10−3 |
| $$= | $$[−1800πcos(πt6)]1037 | |
| $$= | $$17(−1800πcos5π3+1800πcosπ2) | |
| $$≈ | $$−40.9 |
Hence, over the given time frame the fish population decreased by an average of $$40.9 fish per month.
(c) If the initial population when counting began was $$2400 fish, find the model for $$P(t).
Think: Find the indefinite integral of $$P′(t) and then use the population given to solve for the constant of integration.
Do:
| $$P(t) | $$= | $$∫P′(t) dt |
|
| $$= | $$∫300sin(πt6) dt |
|
|
| $$= | $$−1800πcos(πt6)+C |
|
Find $$C using the fact that when $$t=0, $$P(t)=2400.
| $$2400 | $$= | $$−1800πcos(0)+C |
|
| $$∴ C | $$= | $$2400+1800π |
|
Hence, the model for the population of fish after $$t months is: $$P(t)=−1800πcos(πt6)+2400+1800π
Example 2
The marginal profit from the sale of the $$xth item is given by $$P′(x)=0.75x2+2x−6, where $$P(x) is the profit from selling $$x items.
(a) Given that the company incurs a loss of $$$400 if no items are sold, find an expression for $$P in terms of $$x.
Think: We have been given the rate of change in profit per item. We can find the indefinite integral of $$P′(x) and then use the initial profit given to solve for the constant of integration.
Do:
| $$P(x) | $$= | $$∫P′(x) dx |
|
| $$= | $$∫0.75x2+2x−6 dx |
|
|
| $$= | $$0.25x3+x2−6x+C |
|
Find $$C using the fact that when $$x=0, $$P(x)=−400.
| $$−400 | $$= | $$0.25(0)3+(0)2−6(0)+C |
|
| $$∴ C | $$= | $$−400 |
|
Hence, the profit from selling $$x items is given by $$P(x)=0.25x3+x2−6x−400
(b) Hence, determine the profit from selling $$20 items.
Think: We want to find the value of $$P(20).
Do:
| $$P(20) | $$= | $$0.25(20)3+(20)2+6(20)−400 |
|
| $$= | $$1880 |
|
Thus, the profit from selling $$20 items is $$$1880.
(c) Find the net change in profit if the number of items sold changes from $$10 to $$50 items.
Think: We can use the profit function we now have to find the profit for $$10 and $$50 items and then find the difference or equivalently, we can find the integral of $$P′(x) over the given interval.
Do:
| Net change | $$= | $$∫5010(0.75x2+2x−6)dx |
| $$= | $$[0.25x3+x2−6x]5010 | |
| $$= | $$33450−290 | |
| $$= | $$33160 |
Thus, the net change in profit from selling $$10 to $$50 items is $$$33160.
Practice questions
Question 1
An object is cooling and its rate of change of temperature, after $$t minutes, is given by $$T′=−10e−t5.
Determine the instantaneous rate of change of the temperature after $$8 minutes. Give your answer correct to two decimal places where appropriate.
Determine the total change of temperature after $$9 minutes. Give your answer correct to two decimal places.
Hence, determine the average change in temperature over the first $$9 minutes.
Give your answer correct to two decimal places where appropriate.
Question 2
The rate of flow of water into a supply tank is given by $$V′=2000−30t2+5t3, for $$0≤t≤7, where $$V is the amount of water (in litres) in the tank $$t hours after midnight.
Determine the initial flow rate.
Complete the table of values.
$$t $$−1 $$0 $$1 $$4 $$6 $$V′′ $$ $$ $$ $$ $$ Hence state the time $$t when the flow rate is a maximum, and the maximum flow rate at this time.
Maximum flow rate occurs at $$t=
Maximum flow rate = $$ litres/hour
Which of the following is the graph of $$V′?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DFind the area bound by the graph of $$V′ and the $$t-axis between $$t=0 and $$t=3.
What does the area found in part (f) represent?
The average flow rate in the first $$3 hours.
AThe amount of water in the tank at $$3am.
BThe average amount of water in the tank in the first $$3 hours.
CThe total change in the amount of water in the tank in the first $$3 hours.
D
Question 3
The marginal cost for the production of the $$xth item is modelled by $$C′=8x+909, in dollars per item.
Determine the net change in cost for producing between $$13 and $$19 items.
Determine the average change in cost for producing between $$13 and $$19 items.
Question 4
The total revenue, $$R (in thousands of dollars), from producing and selling a new product, $$t weeks after its launch, is such that $$dRdt=401+500(t+1)3.
Determine the revenue function $$R in terms of $$t, expressing your answer in positive index form.
Use $$C as the constant of integration.
Given that the initial revenue at the time of launch was $$0, solve for the constant $$C and hence state the revenue function.
Determine the average revenue earned over the first $$5 weeks. Round your answer to 2 decimal places.
Determine the revenue earned in the $$6th week.