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**preview**shows pages 1-2. to view the full**6 pages of the document.**MA 141 Chapter 4

Section 4.1: Areas and Riemann Sums

To develop the notion of a deﬁnite integral, we will need a special notation for long sums.

This is called summation notation.

n

X

i=1

ai=a1+a2+. . . +an−1+an

The capital Greek letter Σis the symbol for sum and the symbol airepresents the ith element

in the summation. The letter iis called the index of summation. This sum starts at i= 1

and goes up to n. Not all sums start at 1 and other letters could be used for the index of

summation.

Ex: Evaluate the sum

6

X

i=1

3i.

6

X

i=1

3i= 3(1) + 3(2) + 3(3) + 3(4) + 3(5) + 3(6)

= 3 + 6 + 9 + 12 + 15 + 18 = 63

Algebra of Summation: Let nbe a natural number, aiand bibe real numbers for ifrom

1 to n, and c∈R. Then

1)

n

X

i=1

(ai±bi) =

n

X

i=1

ai±

n

X

i=1

bi

2)

n

X

i=1

c·ai=c

n

X

i=1

ai

Ex: Evaluate the sum

4

X

j=1

(2j2+ 3).

4

X

j=1

(2j2+ 3) = 2

4

X

j=1

j2+

4

X

j=1

3

= 2[(1)2+ (2)2+ (3)2+ (4)2] + 3 + 3 + 3 + 3

= 2[1 + 4 + 9 + 16] + 12 = 2(30) + 12

= 72

1

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Figure 1

Now suppose we wanted to ﬁnd the area of

a space bounded by a curve y=f(x), the

x-axis, and the lines x=aand x=b, as

in the image to the left.

There is no simple formula we can use. Instead, we will approximate the area.

The simplest shape we can use to approximate is a rectangle. We will slice the region into

rectangular strips and add up the areas. The thinner our strips, the more accurate the

estimation will be.

Ex: Let f(x) = 16 −x2. Using two rectangles, estimate the area between the curve and the

x-axis on the interval [1,3].

If we use two rectangles, we want them to be the same width, so we will split the interval

into two - [1,2] and [2,3], both of width 1. How should we draw in the rectangles? We have

three basic choices:

Figure 3

im

h

f

3.

w

t

We could draw in rectangles using

the function value at the left-hand

endpoint of each interval. This is

called a left sum.

A1=f(1) ·1 = (16 −12)·1 = 15

A2=f(2) ·1 = (16 −22)·1 = 12

AL=A1+A2= 27

Figur

t

We could draw in rectangles using

the function value at the right-hand

endpoint of each interval. This is

called a right sum.

A1=f(2) ·1 = (16 −22)·1 = 12

A2=f(3) ·1 = (16 −32)·1 = 7

AR=A1+A2= 19

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