Aim: What are Riemann Sums?
description
Transcript of Aim: What are Riemann Sums?
Aim: Riemann Sums & Definite Integrals Course: Calculus
Do Now:
Aim: What are Riemann Sums?
Approximate the area under the curve y = 4 – x2 for [-1, 1] using 4 inscribed rectangles.
Aim: Riemann Sums & Definite Integrals Course: Calculus
Devising a Formula•Using left endpoint to approximate area under the curve is
0 1 2 1nb a
y y y yn
4
3.5
3
2.5
2
1.5
1
0.5
1 2
f x = x2
lower sum
xa b
12
yn - 1yn - 2y0
y1
yn - 1
the more rectangles the better the
approximation
the exact area?
take it to the limit!
0 1 2 1lim nn
b a y y y yn
left endpoint formula
Aim: Riemann Sums & Definite Integrals Course: Calculus
•Using right endpoint to approximate area under the curve is
Right Endpoint Formula
4
3.5
3
2.5
2
1.5
1
0.5
1 2
f x = x2
x
upper sum
a b
y0
y1
yn - 1
yn
1 2 3 nb a
y y y yn
1 2 3lim nn
b a y y y yn
right endpoint formula
1 3 5 2 12 2 2 2
lim nn
b ay y y y
n
midpoint formula
Aim: Riemann Sums & Definite Integrals Course: Calculus
Sigma Notation
sigma
sum of terms
The sum of the first n terms of a sequence is represented by
n
i 1 2 3 4 n,i 1
a a a a a aL
where i is the index of summation,n is the upper limit of summation, and1 is the lower limit of summation.
99
i 1i 1 2 3 99
L
Aim: Riemann Sums & Definite Integrals Course: Calculus
Summation Formulas
constantanyis,.111
caccan
ii
n
ii
n
i
n
iii
n
iii baba
1 11)(.2
cncn
i
1.3
2)1(...321.4
1
nnni
n
i
6)12)(1(...21.5 222
1
2
nnnni
n
i
4)1(...321.6
223333
1
3
nnni
n
i
Aim: Riemann Sums & Definite Integrals Course: Calculus
Riemann Sums•A function f is defined on a closed interval [a, b].•It may have both positive and negative values on the interval.•Does not need to be continuous.
Δx1 Δx2 Δx3 Δx4 Δx5 Δx6
a = = bx0 x6x1 x2 x3 x4 x5
1x 2x 3x 4x 5x 6x
1
0.5
-0.5
-1
1 2 3
Partition the interval into n subintervals not necessarily of equal length.
a = x0 < x1 < x2 < . . . < xn – 1 < xn = b
- arbitrary/sample points for ith intervalix
Δxi = xi – xi – 1
Aim: Riemann Sums & Definite Integrals Course: Calculus
Riemann Sums
Partition interval into n subintervals not necessarily of equal length.
Δx1 Δx2 Δx3 Δx4 Δx5 Δx6
a = x0 x6 = bx1 x2 x3 x4 x5
1x 2x 3x 4x 5x 6x
- arbitrary/sample points for ith intervalix
1
Riemann Sumn
iP ii
R f x x
0 1 2 1nb a b a b a b a
y y y yn n n n
ci = xi
Aim: Riemann Sums & Definite Integrals Course: Calculus
Riemann Sums
1
Riemann Sumn
iP ii
R f x x
Δx1 Δx2
Δx6
x6 = ba = x0
1x 2x 5x
Δx4
Δxi = xi – xi – 1
Aim: Riemann Sums & Definite Integrals Course: Calculus
Definition of Riemann Sum
Let f be defined on the closed interval [a, b], and let Δ be a partition of [a, b] given by
a = x0 < x1 < x2 < . . . . < xn – 1 < xn = b,
where Δxi is the length of the ith subinterval. If ci is any point in the ith subinterval, then the sum
is called a Riemann sum for f for the partition Δ
11
( ) n
i i i i ii
f c x x c x
largest subinterval – norm - ||Δ|| or |P|
b a
n
equal subintervals – partition is regularb a
xn
regular partition general partition
0 implies n
converse not true implies 0n
Aim: Riemann Sums & Definite Integrals Course: Calculus
Model ProblemEvaluate the Riemann Sum RP for
f(x) = (x + 1)(x – 2)(x – 4) = x3 – 5x2 + 2x + 8 on the interval [0, 5] using the Partition P with partition points 0 < 1.1 < 2 < 3.2 < 4 < 5 and corresponding sample points
1 2 3 4 50.5, 1.5, 2.5, 3.6, 5x x x x x
5
1
1 2 31 2 3
4 54 5
iP ii
R f x x
f x x f x x f x x
f x x f x x
1
Riemann Sumn
iP ii
R f x x
Aim: Riemann Sums & Definite Integrals Course: Calculus
Model Problem
5
1
0.5 1.1 0 1.5 2 1.1
2.5 3.2 2 3.6 4 3.2 5 5 4
iP ii
R f x x
f f
f f f
7.875 1.1 3.125 0.9 2.625 1.2
2.944 0.8 18 1 23.9698
Aim: Riemann Sums & Definite Integrals Course: Calculus
Definition of Definite Integral
If f is defined on the closed interval [a, b] and the limit
exists, the f is integrable on [a, b] and the limit is denoted by
The limit is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration.
0 1lim ( )
n
i ii
f c x
0 1
lim ( ) ( )n b
i i ai
f c x f x dx
Definite integral is a numberIndefinite integral is a family of functions
If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].
1
limn
i in i
f c x
Aim: Riemann Sums & Definite Integrals Course: Calculus
Evaluating a Definite Integral as a Limit
2
1
-1
-2
-3
f x = 2x
1
2Evaluate the definite integral 2x dx
3
ib a
x xn n
32ii
c a i xn
1
2 0 1
2 limn
i ii
x dx f c x
1
limn
i in i
f c x
1
3 3lim 2 2n
n i
in n
1
6 3lim 2n
n i
in n
ci = xi
Aim: Riemann Sums & Definite Integrals Course: Calculus
Evaluating a Definite Integral as a Limit1
21
6 32 lim 2n
n i
ix dx
n n
16 3lim 22n
n nn
n n
9lim 12 9 3n n
The Definite Integral as Area of Region
If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded
by the graph of f, the x-axis and the vertical lines x = a and x = b is given by
Area = b
af x dx
2
1
-1
-2
-3
f x = 2x
not the area
cncn
i
1.3
1
( 1)4.2
n
i
n ni
Aim: Riemann Sums & Definite Integrals Course: Calculus
Properties of Definite Integrals
1. If is defined at , then 0
2. If is integrable on [ , ],
then
If is integrable on the three closed intervalsdetermined by , , and , then
3.
a
a
a b
b a
b c c
a a b
f x a f x dx
f a b
f x dx f x dx
fa b c
f x dx f x dx f x dx
If and are integrable on [ , ] and is constant, then the functions of and are integrable on [ , ], and
4.
5. ( )
b b
a a
b b b
a a a
f g a b kkf f g
a b
kf x dx k f x dx
f x g x dx f x dx g x dx
Aim: Riemann Sums & Definite Integrals Course: Calculus
6
4
2
5
g x = x+2
Areas of Common Geometric Figures
2 2
24 x dx
2
Sketch & evaluate area region using geo. formulas.3
14 dx
3
02x dx= 8
1 212
A b b
212
2A r
2
6
4
2
5
f x = 4
A = lw
3
11
1
4 lim
3 1lim 4
2lim 4
lim 4 2 8
n
i in i
n
n i
n
n
dx f c x
n
nn
Aim: Riemann Sums & Definite Integrals Course: Calculus
Model Problems
0
3
Evaluate sin
2
x dx
x dx
=0
3
02x dx
212
6
4
2
5
g x = x+2
2 2
1Evaluate ( 1)x dx
Aim: Riemann Sums & Definite Integrals Course: Calculus
Model Problem
3 2
1
3 3 32
1 1 1
Evaluate 4 3 using each
of the following values.26 , 4, 23
x x dx
x dx x dx dx
3 2
1
3 3 32
1 1 1
4 3
4 3
x x
x dx x dx dx
3 3 32
1 1 1= 4 3x dx x dx dx
26 4= 4 4 3 23 3
Aim: Riemann Sums & Definite Integrals Course: Calculus
Model Problem
3 2
1Evaluate 2 8x dx
15
10
5
-5
-10
2 4
f x = 2x2-8
2
1lim 2 8
n
in i
x x
A1
A2
Total Area = -A1 + A2
41 1ii
c i xn
0 1
lim ( )n
i ii
f c x
4i
b ax x
n n
2
1
4 42 1 8n
i
in n
Aim: Riemann Sums & Definite Integrals Course: Calculus
Model Problem15
10
5
-5
-10
2 4
f x = 2x2-8
A1
A2
2
1
4 42 1 8n
i
in n
2
21
8 16 42 1 8n
i
i in n n
2
21
16 32 46n
i
i in n n
2
2 31
24 64 128n
i
i in n n
2
2 31 1 1
24 64 128n n n
i i i
i in n n
Aim: Riemann Sums & Definite Integrals Course: Calculus
Model Problem
2
2 31 1 1
24 64 128n n n
i i i
i in n n
22 3
1 1 1
24 64 1281n n n
i i i
i in n n
2
1 128 3 124 32 1 26n n n
2
1 128 3 1lim 24 32 1 26n n n n
128 4024 323 3
take the limit n
Aim: Riemann Sums & Definite Integrals Course: Calculus
Definition of Riemann Sum
Aim: Riemann Sums & Definite Integrals Course: Calculus
Definition of Riemann Sum
Aim: Riemann Sums & Definite Integrals Course: Calculus
Subintervals of Unequal Lengths
( ) for the -axis for 0 1f x x x x
1
lim ( )n
i in i
f c x
2
1 2 and is the
width of the th interval
ix x
ni
22
1 2 2
1iix
n n
1.4
1.2
1
0.8
0.6
0.4
0.2
0.5 1
1nn
1n
2n
2
1n
22
2 2
12 nn n
2 2
1 2
2 1i i ix
n
1 2
2 1ix
n
Aim: Riemann Sums & Definite Integrals Course: Calculus
Subintervals of Unequal Lengths
( ) for the -axis for 0 1f x x x x
1
lim ( )n
i in i
f c x
1.4
1.2
1
0.8
0.6
0.4
0.2
0.5 1
1 2
2 1ix
n
2
2 21
2 1limn
n i
i in n
23
1
1lim 2n
n i
i in
3
1 2 1 11lim 26 2n
n n n n nn
3 2
31
4 3 2lim ( ) lim6 3
n
i in ni
n n nf c x
n