Section 4.4 The Fundamental Theorem of Calculus
Motivating Questions
How can we find the exact value of a definite integral without taking the limit of a Riemann sum?
What is the statement of the Fundamental Theorem of Calculus, and how do antiderivatives of functions play a key role in applying the theorem?
What is the meaning of the definite integral of a rate of change in contexts other than when the rate of change represents velocity?
Preview Activity 4.4.1.
A student with a third floor dormitory window 32 feet off the ground tosses a water balloon straight up in the air with an initial velocity of 16 feet per second. It turns out that the instantaneous velocity of the water balloon is given by
Let
represent the height of the water balloon above ground at time and note that is an antiderivative of That is, is the derivative of Find a formula for that satisfies the initial condition that the balloon is tossed from 32 feet above ground. In other words, make your formula for satisfyWhen does the water balloon reach its maximum height? When does it land?
Compute
and What do these represent?What is the total vertical distance traveled by the water balloon from the time it is tossed until the time it lands?
Sketch a graph of the velocity function
on the time interval What is the total net signed area bounded by and the -axis on Answer this question in two ways: first by using your work above, and then by using a familiar geometric formula to compute areas of certain relevant regions.
Subsection 4.4.1 The Fundamental Theorem of Calculus
Suppose we know the position functionExample 4.4.3.
Determine the exact distance traveled on
where
Fundamental Theorem of Calculus.
If
Activity 4.4.2.
Use the Fundamental Theorem of Calculus to evaluate each of the following integrals exactly. For each, sketch a graph of the integrand on the relevant interval and write one sentence that explains the meaning of the value of the integral in terms of the (net signed) area bounded by the curve.
Find a function whose derivative is
Which familiar function has derivative
What is special about
when it comes to differentiation?Consider the derivative of
Find an antiderivative for each of the three individual terms in the integrand.
Subsection 4.4.2 Basic antiderivatives
The general problem of finding an antiderivative is difficult. In part, this is due to the fact that we are trying to undo the process of differentiating, and the undoing is much more difficult than the doing. For example, while it is evident that an antiderivative ofActivity 4.4.3.
Use your knowledge of derivatives of basic functions to complete Table 4.4.5 of antiderivatives. For each entry, your task is to find a function
given function, |
antiderivative, |
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For the table, you might start by constructing a list of all the basic functions whose derivative you know. For the three definite integrals, be sure to recall the sum and constant multiple rules, which work not only for differentiating, but also for antidifferentiating.
Subsection 4.4.3 The total change theorem
Let us review three interpretations of the definite integral.For a moving object with instantaneous velocity
the object's change in position on the time interval is given by and whenever on tells us the total distance traveled by the object onFor any continuous function
its definite integral represents the net signed area bounded by and the -axis on where regions that lie below the -axis have a minus sign associated with their area.-
The value of a definite integral is linked to the average value of a function: for a continuous function
on its average value is given by
Total Change Theorem.
If
differences in heights on
correspond to net signed areas bounded by
Example 4.4.7.
Suppose that pollutants are leaking out of an underground storage tank at a rate of
Since
Thus, the definite integral tells us the total number of gallons of pollutant that leak from the tank from day 4 to day 10. The Total Change Theorem tells us the same thing: if we let
the number of gallons that have leaked from day 4 to day 10.
To compute the exact value of the integral, we use the Fundamental Theorem of Calculus. Antidifferentiating
Thus, approximately 44.282 gallons of pollutant leaked over the six day time period.
To find the average rate at which pollutant leaked from the tank over
gallons per day.
Activity 4.4.4.
During a 40-minute workout, a person riding an exercise machine burns calories at a rate of
What is the exact total number of calories the person burns during the first 10 minutes of her workout?
Let
be an antiderivative of What is the meaning of in the context of the person exercising? Include units on your answer.Determine the exact average rate at which the person burned calories during the 40-minute workout.
At what time(s), if any, is the instantaneous rate at which the person is burning calories equal to the average rate at which she burns calories, on the time interval
What are the units on the area of a rectangle found in a Riemann sum for the function
Use the FTC.
Recall the formula for
Think carefully about which function tells you the instantaneous rate at which calories are burned.
Subsection 4.4.4 Summary
We can find the exact value of a definite integral without taking the limit of a Riemann sum or using a familiar area formula by finding the antiderivative of the integrand, and hence applying the Fundamental Theorem of Calculus.
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The Fundamental Theorem of Calculus says that if
is a continuous function on and is an antiderivative of thenHence, if we can find an antiderivative for the integrand
evaluating the definite integral comes from simply computing the change in on -
A slightly different perspective on the FTC allows us to restate it as the Total Change Theorem, which says that
for any continuously differentiable function
This means that the definite integral of the instantaneous rate of change of a function on an interval is equal to the total change in the function on
Exercises 4.4.5 Exercises
1. Finding exact displacement.
The velocity function is
displacement =
2. Evaluating the definite integral of a rational function.
The value of
3. Evaluating the definite integral of a linear function.
Evaluate the definite integral
4. Evaluating the definite integral of a quadratic function.
Evaluate the definite integral
5. Simplifying an integrand before integrating.
Evaluate the definite integral
6. Evaluating the definite integral of a trigonometric function.
Evaluate the definite integral
7.
The instantaneous velocity (in meters per minute) of a moving object is given by the function
Determine the exact distance traveled by the object on the time interval
What is the object's average velocity on
At what time is the object's acceleration greatest?
Suppose that the velocity of the object is increased by a constant value
for all values of What value of will make the object's total distance traveled on be 210 meters?
8.
A function
Determine the exact value of the net signed area enclosed by
and the -axis on the intervalCompute the exact average value of
onFind a formula for a function
on so that if we extend the above definition of so that if it follows that
9.
When an aircraft attempts to climb as rapidly as possible, its climb rate (in feet per minute) decreases as altitude increases, because the air is less dense at higher altitudes. Given below is a table showing performance data for a certain single engine aircraft, giving its climb rate at various altitudes, where
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Let a new function called
Determine a similar table of values for
and explain how it is related to the table above. Be sure to explain the units.Give a careful interpretation of a function whose derivative is
Describe what the input is and what the output is. Also, explain in plain English what the function tells us.Determine a definite integral whose value tells us exactly the number of minutes required for the airplane to ascend to 10,000 feet of altitude. Clearly explain why the value of this integral has the required meaning.
Use the Riemann sum
to estimate the value of the integral you found in (c). Include units on your result.
10.
In Chapter 1, we showed that for an object moving along a straight line with position function
More recently in Chapter 4, we found that for an object moving along a straight line with velocity function
Are the βaverage velocity on the interval
11.
In Table 4.4.5 in Activity 4.4.3, we noted that for
Suppose that
and let ComputeExplain why
is an antiderivative of for-
Let
and recall thatExplain why
for and for Now discuss why we say that the antiderivative of
is for all