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.
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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