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]....

...No 1. 2. 3. 4. 5. 6. 7. 8. Code: UCCM1153 Status: Credit Hours: 3 Semester and Year Taught:
Information on Every Subject Name of Subject: Introduction to Calculus and Applications
Pre-requisite (if applicable): None Mode of Delivery: Lecture and Tutorial Valuation: Course Work Final Examination 40% 60%
9. 10.
Teaching Staff: Objective(s) of Subject: • Review the notion of function and its basic properties. • Understand the concepts of derivatives. • Understand linear approximations. • Understand the relationship between integration and differentiation and continuity. Learning Outcomes: After completing this unit, students will be able to: 1. describe the basic ideas concerning functions, their graphs, and ways of transforming and combining them; 2. use the concepts of derivatives to solve problems involving rates of change and approximation of functions; 3. apply the differential calculus to solve optimization problems; 4. relate the integral to the derivative; 5. use the integral to solve problems concerning areas.
11.
12.
Subject Synopsis: This unit covers topics on Functions and Models, Limits and Derivatives, Differentiation Rules, Applications of Differentiation and Integrals.
13.
Subject Outline and Notional Hours: Topic Learning Outcomes 1 L 4 T 1.5 P SL 6.25 TLT 11.75
Topic 1: Functions and Models
• • • • • • Functions Models and curve fitting Transformations, combinations, composition and graphs of...

...Answer
* Discussion
* Share
By second fundamentaltheorem of calculus, we obtain
Question 2:
* Answer
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By second fundamentaltheorem of calculus, we obtain
Question 3:
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* Discussion
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By second fundamentaltheorem of calculus, we obtain
Question 4:
* Answer
* Discussion
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By second fundamentaltheorem of calculus, we obtain
Question 5:
* Answer
* Discussion
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By second fundamentaltheorem of calculus, we obtain
Question 6:
* Answer
* Discussion
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By second fundamentaltheorem of calculus, we obtain
Question 7:
* Answer
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By second fundamentaltheorem of calculus, we obtain
Question 8:
* Answer
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By second fundamentaltheorem of calculus, we obtain
uestion 9:
* Answer
* Discussion
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By second fundamentaltheorem of calculus, we obtain
uestion 10:
* Answer
* Discussion
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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...

...How the calculus was invented?
Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was made by Isaac Newton and Gottfried Leibniz. Publication of Newton's main treatises took many years, whereas Leibniz published first (Nova methodus, 1684) and the whole subject was subsequently marred by a priority dispute between the two inventors of calculus.
Greek mathematicians are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Elea discredited infinitesimals further by his articulation of the paradoxes which they create.
Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes.
Archimedes of Syracuse developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See...

...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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