Introduction to summations, Riemann sums, and definite integrals.

1. Summations
Chapters 4.1 -4.3
By Jose Cortez

2. The Basics
• A summation is the sum of a sequence (a1, a2, a3, a4… an)
• A sequence is a listing of terms
• A series is a sum of the terms
• The basic notation of a summation is:
• ∑f(i)=f(1)+f(2)+f(3)+f(4)+…+f(n)
n
i=1
The ending value
The starting value

3. Four special summations
There are four special summations that were given to us:
1.∑1= 1+1+1+1+1+…+1= n
2.∑i=1+2+3+4+…+n=
𝑛(𝑛+1)
2
3.∑i2= 12+22+32+…+n2=
𝑛(𝑛+1)(2𝑛+1)
6
4.∑i3= 13+23+33+…+n3=
n2(n+1)2
4

4. Riemann Sums
• Riemann sums are used to find the area under a curve of a non-negative
function between the vertical lines x=a and x=b.
• The interval is divided into subintervals which all have equal width. Each
subinterval makes a rectangle from which we can approximate the area
under the curve.
• To get the true area under the curve the subinterval’s width must go to zero
(giving an infinite number of subintervals).

5. • xi represents the width of the ith subinterval
(width of the rectangle)
• When finding the area under a curve xi=
𝑏−𝑎
𝑛
(n
represents the number of subintervals)
• ci is the x value on the ith subinterval [a+ xi(i)]
• f(ci) is the height of the rectangle
• f(ci)xi by multiplying them together you get the
area of a rectangle on the ith subinterval
Riemann Sums
xi
a b

6. Riemann Sums (continued)
• 𝑖=1
𝑛
f(ci)xi is the sum of the approximate area of all the rectangles
• lim
𝑛→∞ 𝑖=1
𝑛
f(𝑐𝑖)𝑥𝑖 give the area under a curve above the x-axis on the
given interval.
This limit is called the definite integral of f(x) over the interval [a,b].
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = to area under a curve

7. Riemann Sum Example
𝑖=1
6
5(2𝑖 − 3)2
The 5 can be pulled out of the integral since it is a constant
5 𝑖=1
6
(2𝑖 − 3)2=5 𝑖=1
6
4𝑖2 − 12𝑖 + 9
5[f(1)+f(2)+f(3)+f(4)+f(5)+f(6)]
5[1+1+9+25+49+81]=830

8. Definite Integral Example
A real-life example of definite integrals being used is in the
calculating the distance traveled from 2 points in time.The
area under the curve is the distance traveled between the
two time points.
t1=1 seconds and t2= 5 seconds
At t1 the velocity is 10 m/s
At t2 the velocity is 20 m/s
The equation is
10
4
𝑥 + 7.5
𝑦 =
10
4
𝑥 + 7.5
0
5
10
15
20
25
0 1 2 3 4 5
ChartTitle
The answer is 60 meters as
the distance traveled.

9. Bibliography
• Khan Academy
• Calculus Notes
• http://lugezi.com/images/
• http://tutorial.math.lamar.edu/Classes/CalcI/AreaProblem.aspx