Areas and DistancesThe Distance Problem: find the distance traveled by an objectduring a certain time period if the velocity of the object varies.Problem: Suppose the odometer on our car
Trang 15 INTEGRALS
5.1 Areas and Distances
5.2 The Definite Integral
Discovery Project: Area Functions
5.3 The Fundamental Theorem of Calculus
5.4 Indefinite Integrals and the Net Change Theorem
(Tích phân bất định và định lý biến thiên toàn phần)
Writing Project: Newton, Leibniz, and the Invention of Calculus
Trang 25.1 Areas and Distances
SUMS AND SIGMA NOTATION
We use the symbol Σ to represent a sum, for example
,
2 1
1
2 1
i
n n
i
i
a a
a a
a a
a a
Trang 35.1 Areas and DistancesEvaluating Sums
1
, 4
) 1
( )
4 (
, 6
) 1 2
)(
1
(
3 2
1 )
3 (
, 2
) 1
(
3 2 1 )
2 (
, 1
) 1 (
2 2
1 3
2 2
2 1
2 1 1
n n
n n
i
n
n n
i n
n n
i
Trang 45.1 Areas and Distances
The Basic Area Problem: find the area of a region S.
S
y = f(x)
Problem: Find the area of a region S lying under the graph y = f(x) ≥ 0
and continuous, above the x – axis and between the vertical lines x = a
Trang 55.1 Areas and Distances
PROCEDURE
(1) Divide [a, b] into n equal subintervals by using division points:
a = x0 < x1 < x2 < … < x n – 1 < x n = b the width of each subinterval is Δx = (b – a)/n, x i = a + iΔx.
S1 S2
S
S i
Trang 65.1 Areas and Distances
(2) Build rectangles with base length Δx and height f(x i ), i = 1, 2, …, n The area of each rectangle is f(x i )Δx.
a x1 x2 x i – 1 x i x n – 1 b
f(x i)
Δx
Trang 75.1 Areas and Distances(3) Add area of all rectangles
Trang 85.1 Areas and Distances
Trang 95.1 Areas and Distances
Remark!
(1) We get the same value if we use left endpoints
(2) A more general expression for the area is
with (sample point) is any point in the i-th subinterval [x i – 1 , x i]
] )
(
) ( )
( [ lim
(
) ( )
( [ lim
Trang 105.1 Areas and Distances
Ex. Find the area under the parabola y = x2, above the x-axis and from x = 0 to x = 1.
Sol. Δx = 1/n, x i = 0 + iΔx = i/n,
n n
n n
21
(
n n
)12
)(
1(
n n n
Trang 115.1 Areas and Distances
The Distance Problem: find the distance traveled by an objectduring a certain time period if the velocity of the object varies
Problem: Suppose the odometer on our car is broken and we want toestimate the distance driven over a 30-second time interval We takespeedometer readings every five seconds and record them in thefollowing table
Time(s) 0 5 10 15 20 25 30
Trang 125.1 Areas and Distances
Solution. During each time interval we can estimate the distance byassuming that the velocity is constant
By taking the velocity at the starting point of each time interval, weobtain an estimate for the total distance travelled:
(25×5) + (31×5) + (35×5) + (43×5) + (47×5) + (46×5) = 1135 ft
Trang 135.1 Areas and Distances
By taking the velocity at the end point of each time interval, we obtain
an estimate for the total distance travelled:
(31×5) + (35×5) + (43×5) + (47×5) + (46×5) + (41×5) = 1135 ft
For more accurate estimation, we need to take velocity readingsevery two seconds, or every second, or every half of second, etc
Trang 145.1 Areas and Distances
The area of each rectangle can be interpreted as a distancebecause the height represents velocity and the width represents time
25
5 10 15 20 25 30 t
v Velocity function
Trang 155.1 Areas and Distances
In general, suppose that an object moves with velocity v = f(t), where
a ≤ t ≤ b and f(t) ≥ 0 We take velocity readings at times t0 = a, t1, …, t n =
b so that the velocity is approximately constant on each subinterval If
these times are equally spaced, then the time between consecutive
readings is Δt = (b – a)/n.
The exact distance d traveled is the following limit
t t
f t
t f d
n
i
i n
( lim
Trang 165.1 Areas and Distances
Conclusion:
(1) The distance traveled is equal to the area under the graph of the
velocity function
(2) Some other quantities of interest in the natural and social
sciences—such as the work done by a variable force or the cardiacoutput of the heart—can also be interpreted as the area under acurve
Trang 175.2 The Definite Integral
Def. Let f(x) be a function defined on [a, b] We divide [a, b] into n equal subintervals of width Δx = (b – a)/n by the division points a = x0, x1,
…x n – 1 , x n = b Let be sample point in each subinterval [x i – 1 , x i] Thenthe definite integral of f from a to b is
provided that this limit exists and gives the same value for all possible
choices of sample points If it does exist, we say that f is integrable
on [a, b].
x x
f dx
x f I
n
i
i n
) (
*
i
x
Trang 185.2 The Definite Integral
Note 1. The symbol is called an integral sign, f(x) is called the
integrand, a – lower limit, b – upper limit of integral Theprocedure of calculating an integral is called the integration, dx – indicate the independent variable is x.
Note 2. The definite integral is a number and does not depend on x.
We could use any letter in the place of x.
) ( )
( )
t f dx
x f
Trang 195.2 The Definite Integral
Note 3. The sum
is called a Riemann sum
Note 4. If f takes on both positive and negative values then
x x
2 1
) ( x dx A A f
Trang 205.2 The Definite Integral
Note 5. There are situations in which it is advantageous to workwith subintervals of unequal width In these cases we have to ensurethat all subinterval widths approach 0 in the limiting process In thiscase the definition of definite integral becomes
Note 6. If f has only a finite number of jump discontinuities on [a, b], then is integrable on [a, b].
x b
a
x x
f dx
x f
I
* 0
) (
Trang 215.2 The Definite Integral
Theorem. If f is integrable on [a, b] then
b
a
x x
f dx
x
f
1
) ( lim
) (
n
a
b i a x i a
x n
a
b
x i
Trang 225.2 The Definite Integral
Ex1. Express the limit as a definite integral
2 lim
(2i – 1)/n 1/n
1
2 1
2 lim
2
0
3 1
n
i n
Trang 235.2 The Definite Integral
Ex2. Evaluate the Riemann sum for f(x) = x3 – 6x, taking the sample points to be right endpoints and a = 0, b = 3, n = 6.
2
0
, 2
1 6
0
x i x
1
3 6
1
3
1
) 24
( 16
1 8
24 2
1 2
1 2
6 2
)
(
i i
9375
3 2
) 1 6 (
6 24 4
) 1 6 ( 6 16
Trang 245.2 The Definite IntegralTHE MIDPOINT RULE
)]
(
) ( )
( [ )
( )
1
n n
i
i b
a
x f x
f x
f x x
x f dx
Trang 255.2 The Definite Integral
Ex3. Use the midpoint rule with n = 5 to approximate dx
x
21
1
10
9 2
5
1 2
1
2 1
, 5
1 5
0 19
10 17
10 15
10 13
10 11
10 5
1 1
Trang 265.2 The Definite Integral
Properties of the definite integral
Trang 275.2 The Definite Integral
function even
) ( ,
) ( 2
) ( )
9
(
function odd
) ( ,
0 )
( )
8
(
, )
( )
( )
7
(
, ) ( )
( then
) ( )
( If )
6
(
, ) ( )
( )
( )
5
(
0
x f dx
x f dx
x f
x f dx
x f
dx x
f dx
x f
dx x g dx
x f x
g x
f
dx x f dx
x f dx
x f
a a
Trang 285.3 The Fundamental Theorem of Cal.
The Fundamental Theorem of Calculus establishes a connectionbetween the two branches of calculus: differential calculus and integralcalculus which are inverse processes
The Fundamental Theorem enabled them to compute areas andintegrals very easily without having to compute them as limits of sums
Trang 295.3 The Fundamental Theorem of Cal.
PART I.
Consider a function defined by an equation of the form
where f is continuous on [a, b] and x varies between a and b If x is a
fixed number, then the integral is a definite number If x varies,
the number also varies and defines a function of x, denoted by g(x).
) ( )
( x
a
dt t f x
g
x
a
dt t
f ) (
x
a
dt t
f ) (
Trang 305.3 The Fundamental Theorem of Cal.
If f is a positive function, then g(x) can be interpreted as the area under the graph of f from a to x, where x can vary from a to b.
Area = g(x)
y = f(t)
t
Trang 315.3 The Fundamental Theorem of Cal.
Ex. Let
(a) Find the values of g(0), g(1), g(2), g(3), and g(4).
(b) Sketch a rough graph of g.
( x
a
dt t f x
Trang 325.3 The Fundamental Theorem of Cal.
(b) As f(t) > 0 for t < 3 we keep adding area for t < 3 and so g is increasing up to x = 3 where it attains a maximum value 4.3 For x > 3,
g decreases because f(t) is negative.
x g
Trang 335.3 The Fundamental Theorem of Cal.
) ( )
( lim )
(
Trang 345.3 The Fundamental Theorem of Cal.
The Fundamental Theorem of Calculus, Part 1
If f is continuous on [a, b], then the function g defined by
where a ≤ x ≤ b is continuous on [a, b] and differentiable on (a, b), and
x
a
dt t f x
g ( ) ( )
).
( )
( )
(
dx
d x
Trang 355.3 The Fundamental Theorem of Cal.
Ex1. Find the derivative of function
x t dt x
g
0
2
1 )
(
Trang 365.3 The Fundamental Theorem of Cal.
Ex2. Consider the Fresnel function appeared in Fresnel’s theory ofdiffraction of light waves
Trang 375.3 The Fundamental Theorem of Cal.
Ex3. Find the derivative of following functions
, )
( )
b (
, )
( )
a (
5
4 2
t
dt e
x x
G
dt e
x F
Trang 385.3 The Fundamental Theorem of Cal.
We have the following two additional formulas
).
( ' )) ( ( )
( ' )) ( ( )
( )
2 (
).
( ' )) ( ( )
( )
1 (
) (
) (
) (
x h x h f x
g x g f dt
t
f dx
d
x g x g f dt
t
f dx
d
x g
x h
x g
Trang 395.3 The Fundamental Theorem of Cal.
Examples
(1) Find the derivative of
(2) Solve the integral equation:
x x
t
dt e x
H
) ( 3
2 )
f
Trang 405.3 The Fundamental Theorem of Cal.
PART II.
The Fundamental Theorem of Calculus, Part 2
If F(x) is any antiderivative of f(x) on [a, b], then we have
).
( )
( )
( )
( x dx F x F b F a f
Trang 415.3 The Fundamental Theorem of Cal.
In the physical terms:
If v(t) is the velocity of an object and s(t) is its position at time t, then s’(t) = v(t) Suppose that the object always move in the positive
direction then the area under the velocity curve is equal to the distancetravelled
) ( )
( )
Trang 425.3 The Fundamental Theorem of Cal.
Ex1 Evaluate
Ex2. Is the result exact?
Ex3. Evaluate
.)23
2 cos
1 lim
Trang 435.3 The Fundamental Theorem of Cal.
From then two fundamental theorems of calculus, it is obviously thatdifferential calculus and integral calculus are inverse processes Eachundoes what the other does
).
( )
( )
( '
), ( )
(
a F b
F dx
x F
x f dt
t
f dx d
Trang 445.4 The Indefinite Int & the Net Change Theo.
Indefinite integrals (for more information see 4.8)
The notation is traditionally used for an antiderivative of f
and is called an indefinite integral, hence
For example, we can write
f ( x ) dx
) ( )
( ' )
( )
3
2
3 3
2
x C
x C
x dx
Trang 455.4 The Indefinite Int & the Net Change Theo.
APPLICATION
The fundamental theorem 2 says that if f is continuous on [a, b] then
where F(x) is any antiderivative of f This means F ’ = f, so the above
equation can be rewritten as
) ( )
( )
( )
Trang 465.4 The Indefinite Int & the Net Change Theo.
We know that F ’(x) represents the rate of change of y = F(x) with respect to x and F(b) – F(a) is the change in y when x changes from a
to b As y can change in both directions (increase or decrease), F(b) – F(a) represents the net change in y Thus the FTC 2 is
Net Change Theorem: The integral of a rate of change is the netchange
) ( )
( )
Trang 475.4 The Indefinite Int & the Net Change Theo.
Some applications
(1) If V(t) is the volume of water in a reservoir at time t, then its
derivative V ’(t) is the rate at which water flows into the reservoir at time t So
is the change in the amount of water in the reservoir between time t1and time t2
) ( )
( )
V dt
t V
Trang 485.4 The Indefinite Int & the Net Change Theo.
(2) If [C](t) is the concentration of the product of a chemical reaction
at time t, then the rate of reaction is the derivative d[C]/dt So
is the change in the concentration of C from time t1 to time t2
) ](
[ ) ](
[
]
[
1 2
2
1
t C t
C
dt dt
C d
Trang 495.4 The Indefinite Int & the Net Change Theo.
(3) If the mass of a rod measured from the left end to a point x is m(x), then the linear density is ρ(x) = m’(x) So
is the mass of the segment of the rod that lies between x = a and x = b.
) ( )
( )
Trang 505.4 The Indefinite Int & the Net Change Theo.
(4) If an object moves along a straight line with position function s(t), then its velocity is v(t) = s’(t), so
is the net change of position, or displacement, of the particle during
the time period from t1 to t2
) ( )
( )
2
1
t s t
s dt
t v
Trang 515.4 The Indefinite Int & the Net Change Theo.
A1
A2
A3v
t
3 2
|
, )
(
A A
A dt
t v
A A
A dt
t v
Trang 525.4 The Indefinite Int & the Net Change Theo.
Ex1 A particle moves along a line so that its velocity at time t is
v(t) = t2 – t – 6 (measured in meters per second).
(a) Find the displacement of the particle during the time period [1, 4].(b) Find the distance traveled during this time period
Trang 535.4 The Indefinite Int & the Net Change Theo.
Solution (a)
This means that the particle moved 4.5 m toward the left
(b) v(t) = t2 – t – 6 = (t – 3)(t + 2) The distance travelled is
2
9 )
6 (
10 6
61 )
6 (
) 6 (
| 6
|
4
3
2 3
1
2 4
1
Trang 545.4 The Indefinite Int & the Net Change Theo.
Ex2 Figure shows the power consumption in a certain city for a day(P is measured in megawatts; t is measured in hours starting at midnight) Estimate the energy used on that day
Trang 555.4 The Indefinite Int & the Net Change Theo.
Solution. Power is the rate of change of energy P(t) = E ’(t) So by
the Net Change Theorem the total amount of energy used on that dayis
Use the Midpoint Rule to estimate the total energy used
) 0 ( )
24 ( )
(
24 0
E E
dt t
Trang 565.5 The Substitution Rule
The method of substitution is the integral version of the chain rule and
Let u = g(x) then du/dx = g’(x), or in differential form, du = g’(x) dx
Thus
))
( ( )
( ' )) ( (
))
( ( )
( )
( ' )
( ' )) ( (
Trang 575.5 The Substitution Rule
Substitution in a definite integral
Suppose that g is a differentiable function on [a, b] that satisfies g(a)
= A and g(b) = B Also suppose that f is continuous on the range of g
Then
) ( )
( ' )) (
f dx
x g x g f
Trang 585.5 The Substitution Rule
Trang 60PETROVIETNAM UNIVERSITY
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