1. Chapter 5 Key Concept: The Definite Integral
Section 5.1 How Do We Measure Distance Traveled?
Section 5.2 The Definite Integral
Section 5.3 The Fundamental Theorem and Interpretations
Section 5.4 Theorems About Definite Integrals
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

2. How Do We Measure Distance Traveled?
Section 5.1
How Do We Measure Distance Traveled?
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

3. distance = velocity × time
Estimating the Distance a Car Travels
A car is moving with increasing velocity. Table 5.1 shows the velocity every two seconds:
We might estimate the distance traveled by assuming a constant velocity in each interval and use the formula
distance = velocity × time
At least how far has the car traveled?
20 · · · · · 2 = 360 feet.
At most how far has the car traveled?ining the data
30 · · · · · 2 = 420 feet.1:
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

4. How Do We Improve Our Estimate?
Table Velocity of car every second
Time (sec) 1 2 3 4 5 6 7 8 9 10
Speed (ft/sec) 20 26 30 34 38 41 44 46 48 49 50
New lower estimate = 20 · · · · · 1
+ 41 · · · · · 1
= 376 feet > 360 feet
New upper estimate = 26 · · · · · 1
+ 44 · · · · · 1
= 406 feet < 420 feet
The difference between upper and lower estimates is now 30 feet, half of what it was before. By halving the interval of measurement, we have halved the difference between the upper and lower estimates.
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

5. Visualizing Distance on the Velocity Graph:
Two-Second Data
To visualize the difference between the two estimates, look at Figure 5.1 and imagine the light rectangles all pushed to the right and stacked on top of each other, giving a difference of 30×2=60.
Figure 5.1: Velocity measured every 2 seconds
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

6. Visualizing Distance on the Velocity Graph:
One-Second Data
To visualize the difference between the two estimates, look at Figure 5.2. This difference can be calculated by stacking the light rectangles vertically, giving a rectangle of the same height as before but of half the width. Its area is therefore half what it was before. Again, the height of this stack is 30, but its width is now 1, giving a difference 30.
Figure 5.2: Velocity measured every second
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

7. If the velocity is positive, the total distance traveled is the area under the velocity curve.
Figure 5.3:
Velocity
measured every
½ second
Figure 5.4:
Velocity measured every
¼ second
Figure 5.5:
Distance traveled is area
under curve
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

8. Left- and Right-Hand Sums
If f is an increasing function, as in Figures 5.8 and 5.9, the left-hand sum is an underestimate and the right-hand sum is an overestimate of the total distance traveled. If f is decreasing, as in Figure 5.10 (next slide), then the roles of the two sums are reversed.
Figure 5.8: Left-hand sums
Figure 5.9: Right-hand sums
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

9. Left- and Right-Hand Sums
Figure 5.10: Left and right sums if f is decreasing
For either increasing or decreasing velocity functions, the exact value of the distance traveled lies somewhere between the two estimates. Thus, the accuracy of our estimate depends on how close these two sums are. For a function which is increasing throughout or decreasing throughout the interval [a, b]: Difference between upper and lower estimates = |f(b)-f(a)|Δt.
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

10. Section 5.2 The Definite Integral
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

11. Sigma Notation
Suppose f(t) is a continuous function for a ≤ t ≤ b. We divide the interval from a to b into n equal subdivisions, and we call the width of an individual subdivision Δt, so
Let t0, t1, t2, , tn be endpoints of the subdivisions. Both the left-hand and right-hand sums can be written more compactly using sigma, or summation, notation. The symbol Σ is a capital sigma, or Greek letter “S.” We write
The Σ tells us to add terms of the form f(ti) Δt. The “i = 1” at the base of the sigma sign tells us to start at i = 1, and the “n” at the top tells us to stop at i = n. In the left-hand sum we start at i = 0 and stop at i = n − 1, so we write
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

12. Taking the Limit to Obtain the Definite Integral
Suppose f is continuous for a ≤ t ≤ b. The definite integral of f from a to b, written
is the limit of the left-hand or right-hand sums with n subdivisions of a ≤ t ≤ b as n gets arbitrarily large. In other words,
and
Each of these sums is called a Riemann sum, f is called the integrand, and a and b are called the limits of integration.
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

13. Computing a Definite Integral
Figure 5.20: Approximating
with n = 2
Figure 5.21: Approximating
with n = 10
Figure 5.22: Shaded area is exact value of
When n = 250, a calculator or computer gives < <
So, to two decimal places, we can say that
The exact value is known to be
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

14. The Definite Integral as an Area
Figure 5.23: Area of rectangles
approximating the area under the curve
Figure 5.24: Shaded area is the definite
integral
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

15. When f (x) Is Not Positive
When f (x) is positive for some x values and negative for others, and a < b:
is the sum of areas above the x-axis, counted positively, and areas below the x-axis, counted negatively.
Figure 5.26: Integral
is negative of shaded area
Figure 5.27: Integral
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

16. More General Riemann Sums
A general Riemann sum for f on the interval [a, b] is a sum of the form
where a = t0 < t1 < · · · < tn = b, and, for i = 1, , n,
Δti = ti − ti−1, and ti−1 ≤ ci ≤ ti
Figure 5.28: A general Riemann sum approximating
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

17. Section 5.3 The Fundamental Theorem and Interpretations
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

18. The Fundamental Theorem of Calculus
Theorem 5.1: The Fundamental Theorem of Calculus
If f is continuous on the interval [a, b] and f(t) = F′(t), then
F(b) − F(a) = Total change in F(t) between t = a and t = b =
In words, the definite integral of a rate of change gives the total change.
Since the terms being added up are products of the form “f(x) times a difference in x,” the unit of measurement for is the product of the units for f(x) and the units for x.
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

19. The Definite Integral of a Rate of Change:
Applications of the Fundamental Theorem
Example 2
Let F(t) represent a bacteria population which is 5 million at time t = 0. After t hours, the population is growing at an instantaneous rate of 2t million bacteria per hour. Estimate the total increase in the bacteria population during the first hour, and the population at t = 1.
Solution
Since the rate at which the population is growing is F′(t) = 2t, we have
Change in population = F(1) − F(0) =
Using a calculator to evaluate the integral,
Change in population =
Since F(0) = 5, the population at t = 1 is given by
Population = F(1) = F(0) +
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

20. Calculating Definite Integrals:
Computational Use of the Fundamental Theorem
Example 5
Compute by two different methods.
Solution
Using left- and right-hand sums, we can approximate this integral as accurately as we want. With n = 100, for example, the left-sum is 7.96 and the right sum is Using n = 500 we learn
The Fundamental Theorem, on the other hand, allows us to compute the integral exactly. We take f(x) = 2x. We know that if F(x) = x2, then F′(x) = 2x. So we use f(x) = 2x and F(x) = x2 and obtain
Notice that to use the Fundamental Theorem to calculate a definite integral, we need to know the antiderivative, F. Chapter 6 discusses how antiderivatives are computed.
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

21. Section 5.4 Theorems About Definite Integrals
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

22. Properties of the Definite Integral
Theorem 5.2: Properties of Limits of Integration
If a, b, and c are any numbers and f is a continuous function, then
In words:
1. The integral from b to a is the negative of the integral from a to b.
2. The integral from a to c plus the integral from c to b is the integral from a to b. (This property holds for all numbers a, b, and c, not just for those satisfying a < c < b.)
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

23. Properties of the Definite Integral
Theorem 5.3: Properties of Sums and Constant Multiples of the Integrand
Let f and g be continuous functions and let c be a constant.
In words:
1. The integral of the sum (or difference) of two functions is the sum (or difference) of their integrals.
2. The integral of a constant times a function is that constant times the integral of the function.
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

24. Area Between Curves
If the graph of f(x) lies above the graph of g(x) for a ≤ x ≤ b, then
Area between f and g
for a ≤ x ≤ b
Example 3
Find the area of the
shaded region between
two parabolas in
figure 5.57 to the right.
Solution
The points of intersection must be determined first. Equating f(x) to g(x) and solving for x gives x = 1 and 3. Then, applying Theorem 5.4:
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

25. Using Symmetry to Evaluate Integrals
If f is even, then
If g is odd, then
Figure 5.58: For an even function, Figure 5.58: For an odd function,
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

26. Comparing Integrals Theorem 5.4: Comparison of Definite Integrals
Let f and g be continuous functions.
1. If m ≤ f(x) ≤ M for a ≤ x ≤ b, then
2. If f(x) ≤ g(x) for a ≤ x ≤ b, then
Figure 5.62: The area under the graph of Figure 5.63: If f(x) ≤ g(x) then
f lies between the areas of the rectangles
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

27. The Definite Integral as an Average
Average value of f
from a to b
How to Visualize the Average on a Graph
The definition of average value tells us that
(Average value of f) · (b − a)
Figure 5.65: Area and average value
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved

28. The Definite Integral as an Average
Example 6
Suppose that C(t) represents the daily cost of heating your house, measured in dollars per day, where t is time measured in days and t = 0 corresponds to January 1, Interpret
Solution
The units for the first expression are (dollars/day) × (days) = dollars. The integral represents the total cost in dollars to heat your house for the first 90 days of 2008, namely the months of January, February, and March. The second expression is measured in (1/days)(dollars) or dollars per day, the same units as C(t). It represents the average cost per day to heat your house during the first 90 days of 2008.
Calculus, 6th edition, Hughes-Hallett et.al., Copyright by John Wiley & Sons, All Rights Reserved