Integrals
ArchimedesDiagram:
Archimedes used the method of exhaustion to find an
approximation to the area of a circle. This is an early example of
integration which led to approximate values of π.
Normal Curve
If X is a random variable and X follows the normal curve, then
the probability that X is between x = a and x = b is the area of
the region below the curve, above the x-axis, and between
x = a and x = b .
Example
Let’s approximate the area below the graph of y = f (x) = 1− x 2
between x = 0 and x =1 .
Upper Sums, Lower Sums, Midpoint Sums
Riemann sum calculator:
https://www.desmos.com/calculator/tgyr42ezjq
Approximatetheareabelowthegraphof
y = f (x) = 1− x 2 for 0 ≤ x ≤1 .
Sums
Example1
Findthesumof1+ 2 + 3+ 4 +...+ 99 +100 .(Gauss)
Example2
1+ 2 + 3+ 4 +...+ (n −1) + n SigmaNotation
1+ 2 + 3+ 4 +...+ (n −1) + n =
n
i=
∑
i =1
n ( n +1) .
2
n
12 + 22 + 32 + 4 2 + ... + k 2 + ...+ n2 = ∑ k 2 = n(n +1)(2n +1) 6
k =1
n
2
2
13 + 2 3 + 33 + 4 3 + ... + k 3 + ...+ n 3 = ∑ k 3 = n (n +1) 4
k =1
Example3
Evaluatethefollowingsums:
50
k 2 ∑
k =1
4
∑
i=0
1 2i
4
(−1)k k ∑
k =1
AlgebraRulesforSums:
Expressthefollowingsumsinsigmanotation:
4 + 9 +16 + 25 + 36 1− 1 + 1 − 1 + ...+ 1 2 3 4
77
Evaluatethesums:
10
k 3 ∑
k =1
5
πk
∑
15
k =1
10
( 3k 2 + 7 ) ∑
k =1
13
c
∑ k =1 n
Let’s approximate the area below the graph of y = f (x) = 1− x 2
between x = 0 and x =1 .