5 Accumulated Change: The Deﬁnite Integralwukedi/Chapter5.pdf · MATH 124 Lecture Notes - chapter 5 * Estimating a Deﬁnite Integral from a Table or Graph Example 3 Values for

MATH 124 Lecture Notes - chapter 5

5 Accumulated Change: The Definite Integral

5.1 Distance and Accumulated Change

* How To Measure Distance Traveled and Visualize Distance on the Velocity Graph

Distance = Velocity× Time

Example 1 Suppose that you travel 30 miles/hour for 2 hours, then 40 miles/hour for 1/2 hour, then 20miles/hour for 4 hours.

(a) What is the total distance you traveled?

(b) Sketch a graph of the velocity function for the trip and represent the total distance traveled on thisgraph.

Example 2 Suppose a car is moving with increasing velocity and suppose we measure the car’s velocityevery two seconds, obtaining the following table.

Time(sec) 0 2 4 6 8 10Velocity (ft/sec) 20 30 38 44 48 50

(a) Give lower and upper estimates for the distance this car has traveled.

(b) On a sketch of velocity against time, represent the lower and upper estimates you have obtained in (a).

Page 1 of 25


Example 3 A car starts moving at time t = 0 and goes faster and faster. Its velocity is shown in thefollowing table. Estimate how far the car travels during the 12 seconds.

Time(sec) 0 3 6 9 12Velocity (ft/sec) 0 10 25 45 75

* Visualizing Distance on the Velocity Graph: Area Under Curve

If the velocity is positive, the total distance traveled is the area under the velocity curve.

Example 4 With time t in seconds, the velocity of a bicycle, in feet per second, is given by v(t) = 5t.How far does the bicycle travel from t = 0 to t = 10?

Page 2 of 25


Example 5 The velocity, v, of an object (in meters/sec) is given in the following graph. Estimate the totaldistance the object traveled between t = 0 and t = 6.

t(sec)

v(m/sec)

1 2 3 4 5 6

10

20

30

40

* Approximating Total Change from Rate of Change

Example 6 A city’s population grows at the rate of 5000 people/year for 3 years and then grows at therate of 3000 people/year for the next 4 years. What is the total change in the population of the city duringthis 7-year period?

Page 3 of 25


Example 7 The rate of sales (in games per week) of a new video game is shown in the following table.Assuming that the rate of sales increased throughout the 20-week period, estimate the total number ofgames sold during this period.

Time(weeks) 0 5 10 15 20Rate of sales (games/week) 0 585 892 1875 2350

Example 8 The following figure shows the rate of change of a fish population. Estimate the total changein the population during this 12-month period.

time (months)

rate(fish per month)

2 4 6 8 10 12

5

10

15

20

25

Page 4 of 25


Example 9 Two cars start at the same time and travel in the same direction along a straight road. Thefollowing figure gives the velocity, v, of each car as a function of time, t. Which car:

(a) Attains the larger maximum velocity?

(b) Stops first?

(c) Travels farther?

t(hr)

v(km/hr)

Car B

Car A

Example 10 Tow cars travel in the same direction along a straight road. The following figure show thevelocity, v, of each car at time t. Car B Starts 3 hours after car A and car B reaches a maximum velocityof 100 km/hr.

(a) For approximately how long does each car travel?

(b) Estimate car A’s maximum velocity.

(c) Approximately how far does each car travel?

t (hr)

v (km/hr)Car A

Car B

Page 5 of 25


5.2 The Definite Integral

* Left- and Right-Hand Sums and Definite Integrals

Let f (t) be a function that is continuous for a ≤ t ≤ b. We divide the interval [a, b] into nequal subdivisions, each of width ∆t, so

∆t =b− a

n.

Let t0, t1, t2, · · · , tn be endpoints of the subdivisions. For a left-hand sum, we use the valuesof the function from the left end of the interval. For a right-hand sum, we use the valuesof the function from the right end of the interval. Actually, we have

Left-hand sum =n−1

∑i=0

f (ti)∆t = f (t0)∆t + f (t1)∆t + · · ·+ f (tn−1)∆t

Right-hand sum =n

∑i=1

f (ti)∆t = f (t1)∆t + f (t2)∆t + · · ·+ f (tn)∆t

The definite integral of f from a to b, written∫ b

af (t)dt,

is the limit of the left-hand or right-hand sums with n subdivisions of [a, b] as n getsarbitrarily large. In other words,

∫ b

af (t)dt = lim

n→∞(Left-hand sum) = lim

n→∞

(n−1

∑i=0

f (ti)∆t

)

and ∫ b

af (t)dt = lim

n→∞(Right-hand sum) = lim

n→∞

(n

∑i=1

f (ti)∆t

).

Each of these sums is called a Riemann sum, f is called the integrand, and a and b arecalled the limits of integration.

Page 6 of 25


* Evaluating Left- and Right-hand sums

Example 1 Use the expressions for left- and right-hand sums given on the previous page and the followingtable.

t 15 17 19 21 23f (t) 10 13 18 20 30

(a) If n = 4, what is ∆t? What are t0, t1, t2, t3, t4? What are f (t0), f (t1), f (t2), f (t3), f (t4)?

(b) Find the left and right sums using n = 4.

(c) If n = 2, what is ∆t? What are t0, t1, t2? What are f (t0), f (t1), f (t2)?

(d) Find the left and right sums using n = 2.

* Computing a Definite Integral

Example 2 Compute∫ 3

1 t2dt and represent this integral as an area.

Page 7 of 25


* Estimating a Definite Integral from a Table or Graph

Example 3 Values for a function f (t) are in the following table. Estimate∫ 30

20 f (t)dt by constructingleft- and right-hand sums with n = 5.

t 20 22 24 26 28 30f (t) 5 7 11 18 29 45

Example 4 Estimate∫ 12

0

1x + 1

dx using a left-hand sum with n = 3.

Page 8 of 25


Example 5 The function f (x) is graphed in the following figure. Estimate∫ 6

0 f (x)dx by constructingleft- and right-hand sums with n = 6. Draw the corresponding rectangles for both left- and right-handsums.

x

y

1 2 3 4 5 6

5

10

15

20

Page 9 of 25


Example 6 Given the graph of y = f (t) in the below. Estimate∫ 6

0 f (t)dt by constructing left- andright-hand sums with ∆t = 2. Draw the corresponding rectangles for both left- and right-hand sums.

t

y

1 2 3 4 5 6 7 8

8

16

24

32

Page 10 of 25


5.3 The Definite Integral As Area

* The Definite Integral as an Area: When f (x) is Positive

When f (x) is positive and a < b:

Area under graph of f between a and b =∫ b

af (x)dx.

Example 1 Find the area under the graph of y = x3 + 2 between x = 0 and x = 2.

Page 11 of 25


Example 2 Using the following graph of y = f (x), find the value of∫ 6

1 f (x)dx.

x

y

1 2 3 4 5 6

1

2

3

* Relationship Between Definite Integral and Area: When f (x) is Not Positive

When f (x) is positive for some x-values and negative for others, and a < b:∫ ba f (x)dx is the sum of the areas above the x-axis, counted positively, and the areas below

the x-axis, counted negatively.

Page 12 of 25


Example 3 For each of the function f (x) graphed below, decide whether∫ 3−3 f (x)dx is positive, negative

or approximately zero.

x-3 3

f (x)

x-3 3

f (x)

x-3 3

f (x)

x-3 3

f (x)

Example 4 Use the following graph of y = f (x) to estimate∫ 5−3 f (x)dx

x

y

-3 -2 -1 0 1 2 3 4 5

-3

-2

-1

1

2

Page 13 of 25


Example 5 Given the graph of y = f (x) in the below.

x

y

-4 -3 -2 -1 0 1 2 3 4 5

1

-1

(a) Find∫ 0−3 f (x)dx.

(b) Find∫ 2

0 f (x)dx.

(c) Find∫ 5

1 f (x)dx.

(d) Find∫ 3−3 f (x)dx.

Example 6 Use the following table to estimate the area between f (x) and the x-axis on the interval0 ≤ x ≤ 20.

x 0 5 10 15 20f (x) 15 18 20 16 12

Page 14 of 25


* Area Between Two Curves

If g(x) ≤ f (x) for a ≤ x ≤ b, then

Area between graphs of f (x)and g(x)for a ≤ x ≤ b =∫ b

a( f (x)− g(x))dx.

Example 7 Use an definite integral to find the area under y = 5 ln(2x) and above y = 3 for 3 ≤ x ≤ 5.

Page 15 of 25


Example 8 Find the area between y = x + 5 and y = 2x + 1 between x = 0 and x = 2.

Example 9 Use a definite integral to find the area enclosed by y = 2 + 8x− 3x2 and y = −22 + 2x.

Page 16 of 25


5.4 Interpretations of the Definite Integral

* The Notation and Units for the Definite Integral

The unit of measurement for∫ b

af (x)dx

is the product of the units for f (x) and the units for x.

If f (t) is a rate of change of a quantity, then the Total change in quantity between t = aand t = b is given by ∫ b

af (t)dt.

Example 1 A bacteria colony initially has a population of 14 million bacteria. Suppose that t hours laterthe population is growing at a rate of f (t) = 2t million bacteria per hour.

(a) Give a definite integral that represents the total change in the bacteria population during the timefrom t = 0 to t = 2.

(b) Find the population at time t = 2.

Example 2 Suppose that C(t) represents the cost per day to heat your home in dollars per day, where t istime measured in days and t = 0 corresponds to January 1, 2010. Interpret

∫ 900 C(t)dt.

Example 3 Interpret∫ 3

1 v(t)dt, where v(t) is velocity in meters/sec and t is time in seconds.

Page 17 of 25


Example 4 A cup of coffee at 90◦ is put into a 20◦ room when t = 0. The coffees’s temperature ischanging at a rate of r(t) = −7(0.9t)◦ per minute, with t in minutes. Estimate the coffee’s temperaturewhen t = 10.

Example 5 A man starts 50 miles away from his home and takes a trip in his car. He moves on a straightline, and his home lies on this line. His velocity is given in the following figure and positive velocities takehim toward home.

t (hours)

velocity(mph)

1 2 3 4 5

60

30

0

-30

-60

(a) Does the man turn around? If so, at what time(s)?

(b) When is he going the fastest? How fast is he going then? Toward his home or away?

(c) When is he closest to his home? Approximately how far away is he then?

(d) When is the man farthest from his home? How far away is he then?

Page 18 of 25


Example 6 The rates of growth of the populations of two species of plants are shown in the followingfigure. Assume that the populations of the two species are equal at time t = 0.

t (years)

new plants per year

1 2

1500

3000

4500

species 1

species 2

(a) Which population is larger after one year? After two years?

(b) How much does the population of species 1 increase during the first two years?

Example 7 The following graph shows the rate of change of the quantity of water in a water tower, inliters per day, during the month of April. If the tower had 12,000 liters of water in it on April 1, estimatethe quantity of water in the tower on April 30.

t (days)

rate (liters/day)

6 12 18 24 30

50

100

150

0

100

-50

-100

Page 19 of 25


5.5 The Fundamental Theorem of Calculus

* The Fundamental Theorem of Calculus

The Fundamental Theorem of CalculusIf F′(t) is continuous for a ≤ t ≤ b, then∫ b

aF′(t)dt = F(b)− F(a).

In words: The definite integral of the derivative of a function gives the total change in thefunction.

Example 1 The graph of a derivative f ′(x) is shown in the following figure.

x

y

1 2 3 4 5 6

1

0

-1

Fill in the table of values for f (x) given that f (3) = 2.

x 0 1 2 3 4 5 6f (x) 2

Page 20 of 25


* Marginal Cost and Change in Total Cost

If C′(q) is a marginal cost function and C(0) is the fixed cost,

Cost to increase production from a units to b units = C(b)− C(a) =∫ b

aC′(q)dq

Total variable cost to produce b units =∫ b

0C′(q)dq

Total cost of producing b units = Fixed cost + Total variable cost

= C(0) +∫ b

0C′(q)dq

Example 2 The total cost in dollars to produce q units of a product is C(q). Fixed costs are $20,000. Themarginal cost is

C′(q) = 0.005q2 − q + 56.

(a) On a graph of C′(q), illustrate graphically the total variable cost of producing 150 units.

(b) Estimate C(150), the total cost to produce 150 units.

(c) Find the value of C′(150) and interpret your answer in terms of costs of production.

(d) Use parts (b) and (c) to estimate C(151).

Page 21 of 25


Example 3 A marginal cost function C′(q) is given in the following figure. If the fixed costs are $10,000,estimate:

(a) The total cost to produce 30 units.

(b) The additional cost if the company increases production from 30 units to 40 units.

(c) The value of C′(25). Interpret your answer in terms of costs of production.

q (quantity)

$/unit

10 20 30 40 50 60

10

20

30

40

50

0

C′(q)

Page 22 of 25


Example 4 The marginal cost C′(q) (in dollars per unit) of producing q units is given in the followingtable.

q 0 100 200 300 400 500 600C′(q) 25 20 18 22 28 35 45

(a) If fixed cost is $10,000, estimate the total cost of producing 400 units.

(b) How much would the total cost increase if production were increased one unit, to 401 units?

Example 5 The marginal cost function of producing q mountain bikes is

C′(q) =600

0.3q + 5.

(a) If the fixed cost in producing the bicycle is $2000, find the total cost to produce 30 bicycles.

(b) If the bikes are sold for $200 each, what is the profit (or loss) on the first 30 bicycles?

(c) Find the marginal profit on the 31st bicycle.

Page 23 of 25


Focus on Theory

* The Second Fundamental Theorem of Calculus

Second Fundamental Theorem of CalculusIf f is a continuous function on an interval, and if a is any number in that interval, thenthe function G defined on the interval by

G(x) =∫ x

af (t)dt

has derivative f ; that is, G′(x) = f (x).

Example 1 Suppose that G(x) =∫ x

3

sin√

tt + 6

dt. Find

(a) G′(x).

(b) G(0).

(c) G(7).

(d) G′(2).

Example 2 Let F(b) =∫ b

0 2xdx.

(a) What is F(0)?

(b) Does the value of F increase or decrease as b increases? Assume b ≥ 0.

(c) Estimate F(1), F(2), F(3).

Page 24 of 25


Example 3 For x =0, 0.5, 1.0, 1.5, and 2.0, make a table of values for I(x) =∫ x

0

√t4 + 1dt.

* Properties of the Definite Integral

Sums and Multiples of Definite IntegralsIf a, b, and c are any numbers and f and g are continuous functions, then∫ c

af (x)dx +

∫ b

cf (x)dx =

∫ b

af (x)dx,∫ b

a( f (x) + g(x))dx =

∫ b

af (x)dx +

∫ b

ag(x)dx,∫ b

ac f (x)dx = c

∫ b

af (x)dx.

Example 4 Let∫ b

a f (x)dx = 8,∫ b

a ( f (x))2dx = 12,∫ b

a g(t)dt = 2, and∫ b

a (g(t))2dt = 3. Find thefollowing integrals.

(a)∫ b

a ( f (x) + g(x))dx

(b)∫ b

a (( f (x))2 − (g(x))2)dx

(c)∫ b

a ( f (x))2dx− (∫ b

a f (x)dx)2

(d)∫ b

a c f (z)dz

Example 5 Given∫ 6−4 f (x)dx = 12,

∫ 6−15 g(x)dx = 5, and

∫ 6−4 g(x)dx = −18. Find the following

integrals.

(a)∫ 6

−4[3 f (x)− 5

6g(x)]dx

(b)∫ −4−15 g(x)dx

Page 25 of 25

Documents

5 Accumulated Change: The Deﬁnite Integralwukedi/Chapter5.pdf · MATH 124 Lecture Notes - chapter 5 * Estimating a Deﬁnite Integral from a Table or Graph Example 3 Values for