Download 6.1-6.2 - Math TAMU

c
Math 171, Spring 2011, Benjamin
Aurispa
6.1 Sigma Notation
The Greek letter,
P
, is used to represent a sum of many terms:
If am , am+1 , . . . , an are real numbers and m and n are integers with m < n, then
n
X
ai = am + am+1 + am+2 + · · · + an
i=m
This form is called sigma notation and the letter i is called the index of summation.
Example: Calculate
5
X
(i2 + i).
i=2
Example: Calculate
200
X
200
X
2
i=1
Example: Write the sum
Example: Evaluate
i=4
2
5
+
3
6
+
25 X
1
i=5
2
4
7
1
−
i−1
i
+
5
8
+ ... +
18
21
i=1
using sigma notation.
Properties of Sums Using Sigma Notation:
(a)
n
X
cai = c
n
X
(ai ± bi ) =
i=m
(b)
n
X
ai
i=m
i=m
n
X
i=m
ai ±
n
X
n
X
bi
i=m
1
a
c
Math 171, Spring 2011, Benjamin
Aurispa
Special Sums (Do not memorize):
(a)
n
X
i=
i=1
(b)
n
X
i2 =
n(n + 1)(2n + 1)
6
i3 =
i=1
(c)
n
X
n(n + 1)
2
i=1
Calculate
n(n + 1)
2
n
X
2
(i − 4)(i + 1)
i=1
Calculate lim
n→∞
n
X
2
i=1
"
2i
+
n n
2i
n
3 #
2
c
Math 171, Spring 2011, Benjamin
Aurispa
6.2 Area
Goal: Suppose we have a function f (x) where f (x) ≥ 0 on the interval [a, b]. We want to be able to find
the area under the curve between x = a and x = b.
We can estimate the area by dividing up the region into intervals and then forming rectangles. The area
can then be approximated by the sum of the areas of these rectangles. The more rectangles, the better the
approximation.
3
c
Math 171, Spring 2011, Benjamin
Aurispa
Method:
1. Determine Width of the Rectangles
Create a partition P of the interval [a, b] by dividing the interval into n smaller subintervals.
The x-values we choose to divide the interval into subintervals are called the partition numbers and denoted
x0 , x1 , x2 , . . . , xn .
a
b
. . .
x0
x1
x2
x3
[x0 , x1 ], [x1 , x2 ], [x2 , x3 ], [x3 , x4 ]
...
x4
x n−2
x n−1
xn
[xn−2 , xn−1 ], [xn−1 , xn ]
Notes:
The first partition number should always be a and the last partition number should always be b. However,
the subintervals do not have to be equally spaced.
The lengths of these subintervals will be the widths of our rectangles.
The length of the ith subinterval is denoted ∆xi , where ∆xi = xi − xi−1
The norm of the partition, ||P ||, is defined to be the largest ∆xi . ||P || = max{∆x1 , ∆x2 , . . . , ∆xn }.
4
6
9
14
17
2. Determine Height of the Rectangles
Choose a number within each subinterval [xi−1 , xi ]. We will call this number x∗i . This number can be the
left endpoint, right endpoint, midpoint, or any other point in the subinterval.
We choose the function value at this point, f (x∗i ), to be the height of the rectangle over that interval.
3. Determine Area of the Rectangles
The area of the rectangle corresponding to the subinterval [xi−1 , xi ] is now f (x∗i )∆xi .
So, the total area of all the rectangles is
n
X
f (x∗i )∆xi . This is called a Riemann Sum.
i=1
4
c
Math 171, Spring 2011, Benjamin
Aurispa
Example: Consider the function f (x) = 20 − x2 on the interval [0, 4]. Approximate the area under the
curve on this interval by using the partition P = {0, 2, 3, 4} and choosing x∗i to be the left endpoint of each
subinterval.
If we want n EQUALLY-SPACED subintervals for an interval [a, b], what is ∆xi ?
Example: Consider the function f (x) = x2 + 1 on the interval [2, 6]. Approximate the area under the curve
on this interval by using 4 equal-length subintervals and choosing x∗i to be the midpoint of each subinterval.
5
c
Math 171, Spring 2011, Benjamin
Aurispa
Example: Consider the function f (x) = x3 on the interval [0, 3]. Approximate the area under the curve on
this interval by using 6 equal-length subintervals and choosing x∗i to be the right endpoint of each subinterval.
EXACT Area: These Riemann sums are just an approximation for the area under the curve. As the the
widths of these rectangles get smaller and smaller (and we thus have more and more rectangles), we will get
closer and closer to the actual area.
n
A = lim
||P ||→0
X
f (x∗i )∆xi
i=1
If the intervals all have the same length, this limit can be expressed as
A = lim
n→∞
n
X
f (x∗i )∆xi
i=1
Set up the limit to find the exact area under the graph of f (x) = x3 + 1 on the interval [1, 4] by using equal
subintervals and taking x∗i to be the right endpoint of each subinterval.
6
c
Math 171, Spring 2011, Benjamin
Aurispa
In general then, the exact area under the graph of a curve f (x) on an interval [a, b] can be found by computing
the limit:
n
n
X
X
i(b − a) b − a
∗
lim
f (xi )∆xi = lim
f a+
n→∞
n→∞
n
n
i=1
i=1
Note: We could have also used the left endpoint or midpoint instead of the right endpoint, but it ultimately
doesn’t make a difference since we are taking the limit.
Set up the limit to find the exact area under the graph of f (x) = x2 + x on the interval [3, 7].
The following limit represents the area under the graph of a function f (x) from x = a to x = b. Identify
f, a, and b.
lim
n→∞
n
X
5
i=1
n
s
2+
5i
n
4
+3
7