Class notes

Introduction to Definite Integrals

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Course
MATH_V101 (MATH_V101)

Institution
University Of British Columbia (UBC )

These lecture notes describe the basics of what a definite integral is, how to approximate the area under a curve, and provide an introduction to Reimman Sums and its connection to the Fundamental Theorum of Calculus.

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Uploaded on January 8, 2025
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Lecture #1-Introduction to definite Integrals

learning objectives : ·

Interpret the definite integral (af(x)dX ,
add Basic Idea :

understand why definite integrals sometimes gives negative
·

areas

these areas can be
·
*
solve integrals using geometry calculated using geometry

areas of
*this areaneedstoSee

amenoretreane
·

curved shapes can be
approximated by cutting up the area or a recora
into small rectangles/triangles

explain w/ a picture how to approximate area using left-and right Blemann soms
M integral
ab
·

write Riemann sums in sigma notation

understand the definition of definite integral limit of
·

a as a a Riemann Sum.
D

What is anIntegral ? Area under :
curve ex
-
* a

-
a definite integral is an integral where f(x) =
eX >
- upper bound

Grea Yecont
Approximation and summation notation : the bounds are defined = is

f(x) = eX
* we can subdivide the area into
rectangles

Ifnotite morerecnasa nation
..... J
. More on Riemann Sums :
to definite integral
Sbf(x)dx
>
-
approximate the :

We can break the interval [2 , b] n
into subintervals

Of equal width :
*
left approximations
* right approximation

-

yelds an underestimate-yields an overestimate ↓ for Increasing functions
*x =
for Each rectangle has
opposite decreasing
- :
Area of left most rectangle

3
Width
· :
DX
H
-

area of each rectangle ·

Height :
determined by the functions value at a chosen point in each subinterval (f(xi)
Area of right-most rectangle
(ex :
left right midpoints)
eXi Al .
, ,

-ne(n-1)in Area or a rectangle Is = f(xi) :

DX

Approximating Area
:

= (1 + en + e4n e(n 1)(n) The riemann sum is the sum of all these rectangles
-

:
+ e3/n + ... +

Riemann Sum :
Riemann Sum :
S
= f(x

Where Xi is the Chosen Point

Sin sin In each subinterval

1
7174
= . = 1 7191
.

Each rectangle represents approximation
an of the area under the curve

as n /number of subintervals increases :
·

1 7174
.
<
Sjexdx < 1 7191
.

·
XX (width or rectangles) decreases

* note : this could also be done with midpoints
·

approximation becomes more accurate

* better than left

sf(Xinxi) Dx or ranta Fundamental Theorem

As
of Calculus :

n >-
:

* also trapezoids

= +xi nimf(xi)Dx Sf =

* even more accurate

Example W/Right Biemann Sum

'
Approximate) !
exdx with n = 5

1 .
Interval :
[0 , 1]

. Subinterval
2 width :

AX =
10 = 0 2 .

.
3 Right end paints : 0 2 .
,
0 . 4 , 0 6 .

,
0 8 .
,
1 0
.

4 function evaluations
.
:

7 0 2) , 2(0 4)
. .
, f (0 6) .
,
f(0 8) ,
. f(1)

.
5
Approximate
:
Sum

S = 0 .
2(e0
. 2
+
20
.
4 + 20 .
3 + 20 . a+
el)

In viemann sum :

S = 0 2 .
. gi .
02 2

i =
1

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