Lecture 15 The Definite Integral and Area Under a Curve Definite Integral ---The Fundamental Theorem of Calculus (FTC)

Given that the function [pic] is continuous on the interval [pic] Then, [pic]
where F could be any antiderivative of f on a ( x ( b. In other words, the definite integral [pic] is the total net change of the antiderivative F over the interval from [pic]

• Properties of Definite Integrals (all of these follow from the FTC) 1. [pic]4. [pic]
2. [pic]5. [pic]
3. [pic], k is a constant.

The method of substitution and the method of integration by parts can also be used to evaluate a definite integral.

[pic]
Examples
5. Find [pic] 6.Find [pic]
7.Find [pic]8.Find [pic]
Area and Integration

There is a connection between definite integrals and the geometric concept of area. If f(x) is continuous and nonnegative on the interval [pic], then the region A under the graph of f between [pic]has area equal to the definite integral [pic]. [pic],

where [pic]is any antiderivative of [pic].

• Why the Integral Formula for Area Works?

[pic]

Let A(x) denote the area of the region under f between a and x, then

[pic]

In general,

[pic]
[pic]
By the definition of the derivative,

[pic]
[pic]
[pic]
Since A(a) is the area under the curve between x = a and x = a, which is clearly zero. Hence, [pic]= the area under f between a and b.

Note:The fundamental theorem requires that the function [pic] is non-negative over the interval [pic]. If [pic]is negative over the interval [pic], the definite integral, [pic], results in a value that is the negative of the area between [pic]and the x-axis from[pic] In such a case, the area between the x-axis and the curve is the absolute value of the definite integral, [pic]....

...Answer
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By second fundamentaltheorem of calculus, we obtain
Question 2:
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By second fundamentaltheorem of calculus, we obtain
Question 3:
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By second fundamentaltheorem of calculus, we obtain
Question 4:
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By second fundamentaltheorem of calculus, we obtain
Question 5:
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By second fundamentaltheorem of calculus, we obtain
Question 6:
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By second fundamentaltheorem of calculus, we obtain
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By second fundamentaltheorem of calculus, we obtain
Question 8:
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By second fundamentaltheorem of calculus, we obtain
uestion 9:
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By second fundamentaltheorem of calculus, we obtain
uestion 10:
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By second fundamental...

...Calculus
is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integralcalculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamentaltheorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-definedlimit. Calculus has widespread uses in science, economics, and engineering and can solve many problems that algebra alone cannot.
Calculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If quantities are continually changing, we need calculus to study what is going on.
BRANCHES OF CALCULUSCalculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do...

...Mulan is
released, using real Asian American actors’ voices. Each
Asian sighting that breaks through the constricting
stereotypes gives another reason to celebrate.
16. The primary purpose of the passage is to
(A)
(B)
(C)
(D)
(E)
demand an end to regressive industry practices
examine the impact of a modern invention
analyze the causes of a historical phenomenon
recount a difficult and life-altering event
offer a personal view of a cultural development
17. The family’s usual reaction to an “Asian sighting”
(line 3) is best characterized as
(A)
(B)
(C)
(D)
(E)
excitement
shock
respect
anxiety
disdain
18. In line 5, “Real” most nearly means
(A)
(B)
(C)
(D)
(E)
serious
authentic
practical
utter
fundamental
19. The list of vital signs in lines 17-18 suggests that
the parents’ commentary was
(A)
(B)
(C)
(D)
(E)
-24-
innocent and amusing
technical and bewildering
critical and demoralizing
thorough and systematic
contentious and overwrought
23. The observation about “pre-videocassette recorder
days” (lines 49-51) primarily implies that
20. The author’s description in lines 16-22 (“They
liked . . . TV Asian”) suggests that she was
(A)
(B)
(C)
(D)
(E)
(A) an engineering breakthrough has had unfortunate
consequences
(B) a filming technique has improved the quality of
television programming
(C) a technological innovation has made a certain
experience more common
(D) a common piece of...

...INDETERMINATE FORMS AND IMPROPER
INTEGRALS
Deﬁnition. If f and g are two functions such that
lim f (x) = 0
x→a
and
lim g(x) = 0
x→a
then f (x)/g(x) has the indeterminate form 0/0 at a.
sin t
x2 − 9
Illustration.
has the indeterminate form 0/0 at 0 and
has the
t
x−3
indeterminate form 0/0 at 3.
Theorem. (L’Hopital’s Rule) Let f and g be functions diﬀerentiable on
an open interval I, except possibly at the number a on I. Suppose that for
Chapter 2: INDETERMINATE FORMS AND IMPROPER INTEGRALS
1
all x = a in I, g (x) = 0. If lim f (x) = 0 and lim g(x) = 0, and
x→a
f (x)
if lim
=L
x→a g (x)
x→a
then
f (x)
lim
=L
x→a g(x)
The theorem is valid if all the limits are right-hand limits or all the limits
are left-hand limits.
Illustrations.
1. Because lim sin t = 0 and lim t = 0, we can apply L’Hopitals rule and
t→0
t→0
obtain
sin t
lim
t→0 t
cos t
= lim
t→0 1
= 1
Chapter 2: INDETERMINATE FORMS AND IMPROPER INTEGRALS
2
2. Because lim (x2 − 9) = 0 and lim (x − 3) = 0, we can apply L’Hopitals
x→3
x→3
rule and obtain
x2 − 9
lim
x→3 x − 3
2x
x→3 1
= 6
=
Chapter 2: INDETERMINATE FORMS AND IMPROPER INTEGRALS
lim
3
Examples.
1. Given f (x) =
x
, ﬁnd the lim f (x).
x−1
x→0
e
We can use L’Hopital’s rule since lim x = 0 and lim (ex − 1) = 0
x→0
x
x→0 ex − 1
lim
x→0
1
x→0 ex
= 1
=
lim
x3 − 3x + 2
2. Given f (x) =
evaluate...

...Calculus is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamentaltheorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally considered to have been founded in the 17th century by Isaac Newton and Gottfried Leibniz, today calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot.
Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural...

...THE HISTORY OF CALCULUS
The discovery of calculus is often attributed to two men, Isaac Newton and Gottfried Leibniz, who independently developed its foundations. Although they both were instrumental in its creation, they thought of the fundamental concepts in very different ways. While Newton considered variables changing with time, Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property. On the other hand, Newton used quantities x' and y', which were finite velocities, to compute the tangent. Of course neither Leibniz nor Newton thought in terms of functions, but both always thought in terms of graphs. For Newton the calculus was geometrical while Leibniz took it towards analysis.
It is interesting to note that Leibniz was very conscious of the importance of good notation and put a lot of thought into the symbols he used. Newton, on the other hand, wrote more for himself than anyone else. Consequently, he tended to use whatever notation he thought of on that day. This turned out to be important in later developments. Leibniz's notation was better suited to generalizing calculus to multiple variables and in addition it highlighted the operator aspect of the derivative and integral. As a...

...The Simple Ledger: Ledger Accounts
You will now learn the system used to maintain an up to date financial position.
They use an account and ledger.
Account:
Page specially used to record financial changes
There is one account for each different item affecting the financial position. (Bank, equipment, automobile…)
Ledger:
All the accounts together are called the ledger
Group or file of accounts
Used to record business transaction and keep track of the balances in each specific account
If you wanted to know how much cash the business has to write a cheque, you would look in the ledger (Cash Account)
If you wanted to know how much the business owes on a bank loan, you would look in the ledger (Bank Loan Account)
A ledger can be prepared in different ways (cards, looseleaf ledger, computer system)
T-account:
Simple type of account. (A quick and easy way to track what is happening in each account)
Accounting form we use to keep track of the specific balance in an account
Shaped like a “T”
The formal account, the one actually used in business, will be introduced at a later time.
Important Features of Ledger Accounts
1. Each individual balance sheet item is given its own specially divided page with the name of item at the top (for now think of each “T” as a page)
Each of these pages is called an account
You must learn to call each one by name. i.e., cash account, bank loan account, and so on.
2. The dollar figure for each item is...

...SOLUTIONS TO SUGGESTED PROBLEMS FROM THE TEXT PART 2
3.5 2 3 4 6 15 18 28 34 36 42 43 44 48 49 3.6 1 2 6 12 17 19 23 30 31 34 38 40 43a 45 51 52 1 4 7 8 10 14 17 19 20 21 22 26 r’(θ) = cosθ – sinθ 2 2 cos θ – sin θ = cos2θ z’= -4sin(4θ) -3cos(2 – 3x) 2 cos(tanθ)/cos θ f’(x) = [-sin(sinx)](cosx) -sinθ w’ = (-cosθ)e y’ = cos(cosx + sinx)(cosx – sinx) 2 T’(θ) = -1 / sin θ x q(x) = e / sin x F(x) = -(1/4)cos(4x)
(a) dy/dt = -(4.9π/6)sin(πt/6) (b) indicates the change in depth of water (a) Graph at end (b) Max on 1 July; 4500; yes; 1 Jan (c) pos 1 April; neg 1 Oct (d) 0 2 2 2 (a) a cosθ + √l – a sin θ (b) i: -2a cm/sec 2 2 2 ii: -a√2 – a / (√l – a /2 cm/sec
28 36 37 42 52a 52b 1 2 4b 5 8 13 17 26a 29 39 41 1 2 3 17 22 29 36 44a 46 49 2 5 8 10 14 16b 21 25 26 27 5.2 1 6 8 10 14
Sketch at end Sketch at the end
x = 0: not max/min x = 3/7: local max x = 1: local min
4.2
-1/3 g decreasing near x = x0 g has local min at x1 Sketch at end Sketch at end x = 4; y = 57
Max: 20 at x = 1 Min: -2 at x = -1; x = 8 Max: 2 at x = 0; x = 3 Min: 16 at x = -1; x = 2 (a) f(1) local min; f(0), f(2) local max (b) f(1) global min; f(2) global max
Global min = 2 at x = 1, No global max D=C r = 3B/(2A) Sketch at end Sketch at end. x = L/2 x = 2a Min: -2amps; Max: 2 amps
(a) xy + πy /8 (b) x + y + πy/2 (c) x = 100/(4 + π); y = 200/(4 + π)
2
2t / (t + 1) 1 / (x – 1) cosα/sinα (lnx) + 1 e . 1/x 1 -sin (lnt) / t 2 2 / (√1 – 4t ) 1 / t lnt 2 1 / (1 + 2u + 2u ) 0.8 -1 ‹ x ‹ 1 1 / ((ln...

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