5.3 Fundamental Theorem of
calculus
Fundamental theorem of calculus (FTC) is
one of the greatest accomplishments of the
human mind.
FTC forms a bridge between differential
calculus and integral calculus.
FTC changes an integral problem into an
antideriva
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #10
Due: June 20, 2014
NAME:
STUDENT ID:
sin(x2 ) dx.
Question 1. Use the Trapezoid Rule, using n = 4, to approximate
4
0
Question 2. If we use
University of Saskatchewan
Department of 1\-1athematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #7
Due: June 13, 2014
' imnES
STUDENT ID:
Quasfwu I. Evaluate [c054 swirl-5:3 dx.
= {MW a.»sz a a: d)!
lfcca"; maa) 5a; a QB! mm @zhyd
awa
University of Samskatvl 101mm
I:)C])&-I.I"LII1-1]I} of M M110]mildew and Sta-Ltis Cs
IV-Iath 116 (01) 2014 Summer (Q2
Lab Assignment #5
Due: June. 9, 2014
STUDENT ID:
-_._.,u._. _._._
2 _._ _ _.
RES ion i11( IS-:?' = Ix x' x *4
Q .t ,1. F i/VJJLH
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #2
June 4, 2014
NAME:
STUDENT ID: 0 Q
H
Question I. Use the formulas to nd 2 (213 i]. Simplify your answer.
1:1
f\ n q
. . - 3
Z (BFLN
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #9
Due: June 19, 2014
NAME:
STUDENT ID:
Question 1. Find the length of the curve y = ln(cos x) from x = 0 to x = /4.
4
Question 2. Find the are
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #2
June 4, 2014
NAME:
STUDENT ID:
n
2i3 i . Simplify your answer.
Question 1. Use the formulas to nd
i=1
2
n
Question 2. Use the formulas to nd
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #4
Due: June 6, 2014
NAME:
STUDENT ID:
2
Question 1.
If
2
3 (f (x) + g(x) dx = 8.67 and
3
2g(x) = 2.3, use the properties of denite integrals t
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #8
Due: June 16, 2014
NAME:
STUDENT ID:
Question 1. Complete the square and use a trig substitution to evaluate
1
dx.
x2 + 6x + 5
3
Question 2.
University of Saalmtchewan
Department of Mathenmtiub and Statistics
Math 116 (11) 2014 Summer Q2
Lab Assignment #3
June 5, 2014
(SLUT! 8
NAME:
STUDENT ID:
Question I. Use a. Riemann sum with 6 equal subinterws audheft endguintsias sample points to ap
prox
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #1
Due: June 3, 2014
STUDENT ID:
For each of the following, use algebra. or trig identities to change the form of F (I) so that you can nd its
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #1
Due: June 3, 2014
NAME:
STUDENT ID:
For each of the following, use algebra or trig identities to change the form of F (x) so that you can nd
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #7
Due: June 13, 2014
NAME:
STUDENT ID:
Question 1. Evaluate
cos4 x sin3 x dx.
3
Question 2. Evaluate
cos2 x
dx.
sin x
3
Question 3. Evaluate
Q
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #5
Due: June 9, 2014
NAME:
STUDENT ID:
Question 1. Find
x3
dx
x2 + 1
3
Question 2. Find
cos2 x sin3 x dx
3
Question 3. Find the area of the reg
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #3
June 5, 2014
NAME:
STUDENT ID:
Question 1. Use a Riemann sum with 6 equal subintervals and left endpoints as sample points to approximate th
University of Saskatchewan
Department. of Mathen'mtics and Statis '05
Math 116 (01) 2014 Summer Q2
Lab Assignment #4
Due: June 61 2014
NAME:
STUDENT ID:
Question I. If 3 + g(:r):} d3: = 8.67 and 29hr] = 2.3, use the prop 'ties of denite integrals to 3
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) 2014 Summer Q2
Lab Assignment #8
Due: June 16, 2014
A = 5040117013
STUDENT ID:
1
Question I. [Evaluate / _-
Vn'2 + 6:3 + 5
d3.
Clima- I; 3 «a 59 Y 5-
. {x74 (22)
Chapter 5 Integrals
This chapter is very important. It will give
you an overall picture of what the course is
like.
There are 3 topics:
Topic 1. Definition of a definite integral
(5.15.2):
Area problems lead to formulate the definition
of a definite integ
5.4 Indefinite integrals and
the net change theorem
I.
Indefinite integrals:
A convenient notation for antiderivatives.
II. Table of indefinite integrals
The table of derivatives in math 110 generates
the table of indefinite integrals.
Must memorize this
Integral Calculus
There are two branches in calculus:
Differential Calculus
Integral Calculus
Math 110 (Differential Calculus):
Given f(x), find f (x) (f is called the derivative of f)
Applications of derivatives
Math 116 (Integral Calculus):
Given f(x),
5.2 The definite integral
I. Definition of a definite integral
Tangent line and velocity problems lead us to
formulate the notion of derivative
Area problems lead us to formulate the notion
of definite integral
II. Midpoint rule
another type of Riemann s
5.5 Substitution Rule (SR)
SR is the first basic technique we learn to find
antiderivatives.
SR will be used many times throughout this course.
You have to practice SR a lot.
There is no one specific technique that applies to all
different functions. We
Ch 6 Applications of integration
This chapter explores some applications of
integration.
In applications, the key is to set up a
definite integral for the given quantity.
1
A general approach to set up an integral for a quantity Q :
(1) Put Q in a coordin
6.3 Volumes By Cylindrical Shell
In disk-washer method:
slices the rotation axis
rotation axis is the the integration variable.
In cylindrical shell method:
slices of the region / the rotation axis
the other axis as integration variable, i.e.,
integrate
6.4 Work
Topics:
I. General Idea
II. Work related to spring
III.Work related to pumping water and others
1
Recall:
In applied problems, we follow the following general
approach to set up the definite integral for the quantity
Q to be found
Put Q in a coo
6.5 Average value of a function
We know how to compute the average value
for finite many numbers
How to compute the average value for a
continuous function over an interval ?
Meteorologists frequently speak of average
temperature. How to compute the aver
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) Summer 2014 Q2
June 10, 2014 Midterm Test # 1 80 minutes
This is a formal assessment. Cheating on an assessment is considered a serious offense by the University and can be
University of Saskatchewan
Department of Mathematics and Statistics
Math 116 (01) Summer 2014 Q2
June 17, 2014 Midterm Test # 2 80 minutes
This is a formal assessment. Cheating on an assessment is considered a serious oense by the University and can be
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