Chapter 1
The fundamental theorem of
Calculus
In Dierential Calculus, we have learned about derivatives of functions, and their uses. Integral Calculus,
as we shall see, uses the reverse concept, i.e. the concept of an anti-derivative.
As in the case of D
72
CHAPTER 2. TOOLS FOR INTEGRATION
2.4
Integration by parts
Integration by parts is the next most used technique for integration after substitution. Just as the substitution method was a consequence of the Chain Rule for derivatives, integration by parts
92
CHAPTER 2. TOOLS FOR INTEGRATION
2.7
2.7.1
Improper integrals
Case Study: Revisiting probability distributions
In many of the Case Studies we have looked at so far, we encountered probability distributions functions
where p(x) goes to 0 as x tends to i
87
2.6
2.6.1
Integrals of functions with radicals, and the length of curves.
Case Study: The Golden Gate Bridge
The Golden Gate Bridge is a remarkable engineering feat and one of the most beautiful bridges in the
world. It was designed in the ealy 30s and
Chapter 3
Advanced tools for the solution of
ODEs
3.1
3.1.1
Numerical methods
Case Study: The SIR epidemic model
One of the main applications of Calculus in biology and health sciences is in epidemiology, which models
the propagation of epidemics given so
AMS 15B, Midterm 1
Name:
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Read all the questions before you start working on any of them. Start with the ones you are most
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AMS 15B, Midterm 2
Name: ____________________________ __
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AMS 15B, Midterm 2
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80
2.5
CHAPTER 2. TOOLS FOR INTEGRATION
Integrals of trigonometric functions
It is very common in various problems in physics and engineering to come across integrals of trigonometric
functions. In this short case study, we will look at a very topical app
64
2.3
CHAPTER 2. TOOLS FOR INTEGRATION
Integrating rational functions
In the Logistic Equation Case Study, we ended up having to integrate a rational function. Many problems,
in fact, lead to integrations of rational functions. This section will guide yo
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Homework 6: Improper integrals and more ODE problems
In ALL questions, you MUST justify your answers step by step. Missing steps will not count for full
credit.
1 Integrals of odd and even functions
In all the following cases, dete
13
In this Case Study, we will continue to look at Antiderivatives as solutions of dierential equations.
In this case, however, we will see a dierent class of dierential equations which are called autonomous
and occur commonly in studies of population dyn
33
Example: What is
a
0
x2 dx?
Note that, in practice, integrals are RARELY calculated using Riemann sums. (just as derivatives
were rarely ever calculated using the formal denition of derivatives). More on this later.
Graphical interpretation of definite
23
1.2
Integrals
In this Section, we change tack completely and look at an apparently unrelated problem, which deals with
probabilities. We will learn about discrete probabilities, sums, and learn about the summation convention.
Then we will move to conti
37
1.3
The fundamental theorem of Calculus
In the nal section of this Chapter, we will study the relationship between antiderivatives and integrals,
to discover that they are intimately related through the Fundamental Theorem of Calculus.
1.3.1
Case Study
58
CHAPTER 2. TOOLS FOR INTEGRATION
2.2
Application of the method of substitution: ODEs
In this Section, we will re-visit ordinary dierential equations and re-interpret the method of solution
using integrals (instead of antiderivatives). The method of sub
53
2.1.8
Mathematical corner: The method of substitution (part II)
The substitution method learned above can be generalized to transform variables x into u even when u is
x
not a linear function of x. The principle is very much the same. Given an integral
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Chapter 2
Tools for integration
In the last Chapter, we learned a number of general concepts related to integration and saw that integrals
and derivatives are inverse operation of one-another. Thanks to that, we can now calculate a fairly large
number of
AMS 15B, Practice Final
Name:
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AMS 15B, Midterm 1
Name:
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Homework 3: The Fundamental Theorem of Calculus, Method of
Substitution
1
Calculus problems [75 points]
In ALL questions, you MUST justify your answers step by step. Missing steps will not count for full
credit.
1.1
The fundamental theorem of Calculus
Eva
Homework 4: ODEs, Rational functions and integration by parts.
1
Calculus problems [75 points]
In ALL questions, you MUST justify your answers step by step. Missing steps will not count for full
credit.
1.1
Revisiting ODEs
Find the general, and then the s
Homework 5: Miscellaneous integration techniques
In ALL questions, you MUST justify your answers step by step. Missing steps will not count for full
credit.
1 Miscellaneous integration techniques
Evaluate the following integral using any technique of y
AMS 1513 Final 2012
Name: __L_.M,,,,,,L_
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Read all the questions before you start working on any of them. Start with the ones you are most
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