MAT 136 Calculus I Lecture Notes
Li Chen
November 15, 2018
Contents
1 How to Use These Notes 1
2 Introduction to the Integral 2
2.1 (5.1) Area and Distances . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.2 Displacement/Distance . . . . . . . . . . . . . . . . . . . 3
2.1.3 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 (5.2) The Definite Integral . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Definition of the Definite Integral . . . . . . . . . . . . . . 4
2.2.2 Properties of the Sigma Sum . . . . . . . . . . . . . . . . 5
2.2.3 Properties of the Integral . . . . . . . . . . . . . . . . . . 5
2.3 (4.9) Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 (5.3) Fundamental Theorem of Calculus (FTC) . . . . . . . . . . 6
2.5 (5.4) Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . 7
3 Techniques of Integration 9
3.1 (5.2) Linearity [? ? ? ? ?] . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 (5.5) The Substitution Rule . . . . . . . . . . . . . . . . . . . . . 10
3.3 (7.1) Integration by Parts . . . . . . . . . . . . . . . . . . . . . . 14
3.4 (7.1) Trig Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Trig Substitution 19
5 Other Techniques 19
1 How to Use These Notes
First let me state the two most important questions we want to answer in this
class
How to integrate a function f (x)
1
,The majority of the contents in this note is geared to answer this questions.
In these notes, I will explicitly indicate which are the important materials.
Although one should be comfortable with all the materials in this note, but
certain critical concept must be emphasized to illuminate the essence of the
course. First, definitions are in bold. I will use a scale of stars to measure the
importance of the material. The importance of the concept decreases as the
number of star decreases. Whenever you see a 5-star tag [? ? ? ? ?], this means
you must know this inside out. And [?] is just a bit more important than the
regular text you see in the notes. Finally, I will italicize item that 1) you need
to be careful with as they might be confusing, or 2) to leave a comment.
The corresponding section numbers in [Stewart] are listed in bracket in the
heading of each section. And I will NOT write the integration constant +C in
the notes to save some ink, but you should always write it!!
2 Introduction to the Integral
2.1 (5.1) Area and Distances
2.1.1 Area
A mathematical illustrative example of the integral is area under a curve. Let
f (x) be a non-negative continuous function. We will say the area under y =
f (x) from x = a to x = b to be the area bounded between the lines x = a,
x = b, the x-axis, and the graph of f (x).
y
3
2 f (x) = x2
1
0 x
−4 −3 −2 −1 0 1 2 3 4
−1
−2
−3
2
, In this example, the shaded region represents the area under the curve y =
f (x) = x2 from x = −2 to x = 2. In general, to find the area under the curve
y = f (x) from x = a to x = b, we divide the interval [a, b] into segments
[x0 , x1 ], [x1 , x2 ], · · · , [xn−1 , xn ] (2.1.1)
of width ∆x. That is,
b−a
∆x = xi − xi−1 = (2.1.2)
n
and xi = a + i∆x for all i = 1, ..., n. On each interval, the funtion f (x) ≈ f (xi )
since the interval is very small so that f (x) is about constant. So that the area
under f (x) is approximately f (xi )∆x. Hence, we can then approximate the
area by
f (x1 )∆x + f (x2 )∆x + · · · + f (xn )∆x. (2.1.3)
In fact the choice of sample point does not matter. In fact, we can choose any
sample point x∗i in the interval [xi−1 , xi ] and compute the sum
f (x∗1 )∆x + f (x∗2 )∆x + · · · + f (x∗n )∆x. (2.1.4)
As we choose the interval [xi−1 , xi ] to be small, equivalently ∆x to be smaller.
As we choose the the width ∆x to smaller by taking the number of segments
n → ∞, the area under f (x) from x = a to x = b is
A = lim f (x∗1 )∆x + f (x∗2 )∆x + · · · + f (x∗n )∆x. (2.1.5)
n→∞
2.1.2 Displacement/Distance
A physically motivating example for the integral is the displacement traveled
by a car with velocity f (t) at time t. Suppose that from time t = a to t = b a
car travels at a velocity f (t). If f (t) = v is a constant. Then the displacement
traveled in ∆t units of time is simply
d = v∆t = f (t)∆t. (2.1.6)
Now suppose that f (t) is variable. If it is continuous, then its value varies
slightly if t changes by a small amount. Hence we may approximate the distance
traveled by dividing the time interval [a, b] into small segments
[t0 , t1 ], [t1 , t2 ], · · · , [tn−1 , tn ] (2.1.7)
b−a
of width ∆t = n . On each time segment [ti−1 , ti ], the car travels approxi-
mately
f (t∗i )∆t (2.1.8)
3
Li Chen
November 15, 2018
Contents
1 How to Use These Notes 1
2 Introduction to the Integral 2
2.1 (5.1) Area and Distances . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.2 Displacement/Distance . . . . . . . . . . . . . . . . . . . 3
2.1.3 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 (5.2) The Definite Integral . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Definition of the Definite Integral . . . . . . . . . . . . . . 4
2.2.2 Properties of the Sigma Sum . . . . . . . . . . . . . . . . 5
2.2.3 Properties of the Integral . . . . . . . . . . . . . . . . . . 5
2.3 (4.9) Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 (5.3) Fundamental Theorem of Calculus (FTC) . . . . . . . . . . 6
2.5 (5.4) Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . 7
3 Techniques of Integration 9
3.1 (5.2) Linearity [? ? ? ? ?] . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 (5.5) The Substitution Rule . . . . . . . . . . . . . . . . . . . . . 10
3.3 (7.1) Integration by Parts . . . . . . . . . . . . . . . . . . . . . . 14
3.4 (7.1) Trig Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Trig Substitution 19
5 Other Techniques 19
1 How to Use These Notes
First let me state the two most important questions we want to answer in this
class
How to integrate a function f (x)
1
,The majority of the contents in this note is geared to answer this questions.
In these notes, I will explicitly indicate which are the important materials.
Although one should be comfortable with all the materials in this note, but
certain critical concept must be emphasized to illuminate the essence of the
course. First, definitions are in bold. I will use a scale of stars to measure the
importance of the material. The importance of the concept decreases as the
number of star decreases. Whenever you see a 5-star tag [? ? ? ? ?], this means
you must know this inside out. And [?] is just a bit more important than the
regular text you see in the notes. Finally, I will italicize item that 1) you need
to be careful with as they might be confusing, or 2) to leave a comment.
The corresponding section numbers in [Stewart] are listed in bracket in the
heading of each section. And I will NOT write the integration constant +C in
the notes to save some ink, but you should always write it!!
2 Introduction to the Integral
2.1 (5.1) Area and Distances
2.1.1 Area
A mathematical illustrative example of the integral is area under a curve. Let
f (x) be a non-negative continuous function. We will say the area under y =
f (x) from x = a to x = b to be the area bounded between the lines x = a,
x = b, the x-axis, and the graph of f (x).
y
3
2 f (x) = x2
1
0 x
−4 −3 −2 −1 0 1 2 3 4
−1
−2
−3
2
, In this example, the shaded region represents the area under the curve y =
f (x) = x2 from x = −2 to x = 2. In general, to find the area under the curve
y = f (x) from x = a to x = b, we divide the interval [a, b] into segments
[x0 , x1 ], [x1 , x2 ], · · · , [xn−1 , xn ] (2.1.1)
of width ∆x. That is,
b−a
∆x = xi − xi−1 = (2.1.2)
n
and xi = a + i∆x for all i = 1, ..., n. On each interval, the funtion f (x) ≈ f (xi )
since the interval is very small so that f (x) is about constant. So that the area
under f (x) is approximately f (xi )∆x. Hence, we can then approximate the
area by
f (x1 )∆x + f (x2 )∆x + · · · + f (xn )∆x. (2.1.3)
In fact the choice of sample point does not matter. In fact, we can choose any
sample point x∗i in the interval [xi−1 , xi ] and compute the sum
f (x∗1 )∆x + f (x∗2 )∆x + · · · + f (x∗n )∆x. (2.1.4)
As we choose the interval [xi−1 , xi ] to be small, equivalently ∆x to be smaller.
As we choose the the width ∆x to smaller by taking the number of segments
n → ∞, the area under f (x) from x = a to x = b is
A = lim f (x∗1 )∆x + f (x∗2 )∆x + · · · + f (x∗n )∆x. (2.1.5)
n→∞
2.1.2 Displacement/Distance
A physically motivating example for the integral is the displacement traveled
by a car with velocity f (t) at time t. Suppose that from time t = a to t = b a
car travels at a velocity f (t). If f (t) = v is a constant. Then the displacement
traveled in ∆t units of time is simply
d = v∆t = f (t)∆t. (2.1.6)
Now suppose that f (t) is variable. If it is continuous, then its value varies
slightly if t changes by a small amount. Hence we may approximate the distance
traveled by dividing the time interval [a, b] into small segments
[t0 , t1 ], [t1 , t2 ], · · · , [tn−1 , tn ] (2.1.7)
b−a
of width ∆t = n . On each time segment [ti−1 , ti ], the car travels approxi-
mately
f (t∗i )∆t (2.1.8)
3
