The document provides information about sigma notation, approximating the area under a curve using rectangles, and defining the exact area of a plane region. It includes an example of using rectangles with varying widths to better approximate the area under the parabola y=x^2 from 0 to 1. As the number of rectangles increases from 4 to 8 to 50 and 1000, the upper and lower estimates of the area converge, allowing for a more accurate determination of the actual area.

1. 2/7/2017
1
4 INTEGRATION
2
Information 8 February 2017
• Homework on par 4.1 is available on uLink.
• Prepare your homework for the tutorial
class on Thursday.
• Check uLink for all information.
Area4.1
4
content
1. Use sigma notation to
write and evaluate a sum.
2. Understand the concept
of area.
3. Use rectangles to
approximate the area of a
plane region.
4. Find the area of a plane
region using limits.
1
2
3
4
5
Sigma Notation
Given a sequence
a1, a2, a3, a4, . . .
we can write the sum of the first n terms using
SUMMATION NOTATION, or SIGMA NOTATION. This
notation derives its name from the Greek letter  (capital
sigma, corresponding to our S for “sum”).
6
Sigma Notation

2. 2/7/2017
2
7
Additional Example 1
8
Sigma Notation
The following properties of sums are natural consequences of
properties of the real numbers.
9
Additional Theorem
10
10
Sums of Powers of Integers
Sums of Powers of Integers
1
( 1)
1 2 3 41
2
.
n
i
n n
i n


      
2 2 2 2 2 2
1
( 1)(2 1)
1 2 3 42.
6
n
i
n n n
i n

 
      
2 2
3 3 3 3 3 3
1
( 1)
1 2 3 43.
4
n
i
n n
i n


      
2
4 4 4 4 4 4
1
( 1)(2 1)(3 3 1)
1 2 3 44.
30
n
i
n n n n n
i n

   
      
2 2 2
5 5 5 5 5 5
1
( 1) (2 2 1)
1 2 3 4
1
5
2
.
n
i
n n n n
i n

  
      
11
Additional Example 2
12
The Area Problem

3. 2/7/2017
3
13
The Area Problem
We begin by attempting to solve the area problem: Find the area
of the region S that lies under the curve y = f(x) from a to b.
This means that S, illustrated in Figure 1, is bounded by the
graph of a continuous function f [where f(x)  0], the vertical
lines x = a and x = b, and the x-axis.
Figure 1 14
The Area Problem
However, it isn’t so easy to find the area of a region with curved
sides.
We all have an intuitive idea of what the area of a region is. But
part of the area problem is to make this intuitive idea precise by
giving an exact definition of area.
We first approximate the region S by rectangles and then we
take the limit of the areas of these rectangles as we increase
the number of rectangles.
The next example illustrates the procedure.
15
Additional Example
Use rectangles to estimate the area under the parabola y = x2
from 0 to 1 (the parabolic region S illustrated in Figure 3).
Figure 3
16
Additional Example - Solution
Solution:
We first notice that the area of S must be somewhere
between 0 and 1 because S is contained in a square with
side length 1, but we can certainly do better than that.
Suppose we divide S into
four strips S1, S2, S3, and S4
by drawing the vertical lines
, , and as in
Figure 4(a).
Figure 4(a)
17
Example 1 – Solution
We can approximate each strip by a rectangle whose base is
the same as the strip and whose height is the same as the right
edge of the strip [see Figure 4(b)].
cont’d
Figure 4(b) 18
Additional Example – Solution
In other words, the heights of these rectangles are the
values of the function f(x) = x2 at the right endpoints of the
subintervals , , , and .
Each rectangle has width and the heights are
and 12.
If we let R4 be the sum of the areas of these approximating
rectangles, we get
cont’d

4. 2/7/2017
4
19
Additional Example – Solution
From Figure 4(b) we see that the area A of S is less than
R4, so
cont’d
Figure 4(b)
20
Additional Example – Solution
Instead of using the rectangles in Figure 4(b) we could use
the smaller rectangles in Figure 5 whose heights are the
values of f at the left endpoints of the subintervals.
(The leftmost rectangle has collapsed because its height is
0.)
cont’d
Figure 5
21
Additional Example – Solution
The sum of the areas of these approximating rectangles is
We see that the area of S is larger than L4, so we have
lower and upper estimates for A:
We can repeat this procedure with a larger number of
strips.
cont’d
22
Additional Example – Solution
Figure 6 shows what happens when we divide the region S
into eight strips of equal width.
cont’d
(a) Using left endpoints (b) Using right endpoints
Approximating S with eight rectangles
Figure 6
23
Additional Example – Solution
By computing the sum of the areas of the smaller
rectangles (L8) and the sum of the areas of the larger
rectangles (R8), we obtain better lower and upper
estimates for A:
So one possible answer to the question is to say that the
true area of S lies somewhere between 0.2734375 and
0.3984375.
We could obtain better estimates by increasing the number
of strips.
cont’d
24
Additional Example – Solution
The following table shows the results of similar calculations
(with a computer) using n rectangles whose heights are
found with left endpoints (Ln) or right endpoints (Rn).
cont’d

5. 2/7/2017
5
25
Additional Example – Solution
In particular, we see by using 50 strips that the
area lies between 0.3234 and 0.3434.
With 1000 strips we narrow it down even more:
A lies between 0.3328335 and 0.3338335.
A good estimate is obtained by averaging these
numbers:
A  0.3333335
cont’d
26
Additional Example
27
Definition of the Area of a Region in the Plane
28
Additional Example