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Solved Problems on Numerical Integration
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Review of the Subject
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Definite Integrals f t
k( ) Δxk
k=1
n
∑ D→ 0
⏐ →⏐ ⏐ ⏐
f x( )dx
a
b
∫
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
NUMERICAL APPROXIMATIONS
Decompose [a,b] into n subintervals.
Length of a subinterval:
Δx =
b −an
.
kth subinterval:
a + k −1( ) Δx, a + kΔx⎡
⎣⎤⎦.
Riemann sum:
f t
k( ) Δxk=1
n∑ .
Tag-points tk can be chosen freely. a + k −1( ) Δx ≤t
k≤a + kΔx.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Left Approx.
f a + k −1( ) Δx( ) Δx
k=1
n
∑
Right Approx.
f a + kΔx( ) Δx
k=1
n
∑APPROXIMATIONS FOR
f x( )dx
a
b
∫ Δx =
b −an
.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Midpoint Approximation
MID(n) =
Trapezoidal Approximation
TRAP n( ) =
LEFT n( ) +RIGHT n( )
2.
APPROXIMATIONS FOR
f x( )dx
a
b
∫ Δx =
b −an
.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
SIMPSON’S APPROXIMATION
In many cases, Simpson’s Approximation gives best results.
SIMPSON n( ) =
2MID n( ) + TRAP n( )
3.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
PROPERTIES
LEFT(n) ≤ f x( )dx ≤
a
b
∫
If f is increasing,Property
RIGHT(n)
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
COMPARING APPROXIMATIONS
Property
a b
f
LEFT n( ) ≤MID n( ) ≤ f x( )dxa
b
∫≤TRAP n( ) ≤RIGHT n( )
If f is increasing and concave-up,
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problems
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problems
1
x +1 − x −1 dx
−2
2
∫ =?
2
f x( )dx
−3
3
∫ =?
Speed given by table. Estimate the distance traveled.
t (s) 0 1 2 34 5 6 7 89 1
0
s (m/s) 01
428
42
55
67
30
30
45
57
703
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
4
x3 dx
0
10
∫ .
Approximate the value of the integral
Which method gives the best result?
5
e
−x2
2 dx0
2
∫ .
Approximate the value of the integral
Estimate the errors.
Problems
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Integrals from graphs
x +1 − x −1 dx
−2
2
∫ .
Problem Compute the integral
Solution
Draw the graph of the function and compute the integral as the area under the graph.
First get rid of the absolute value signs.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x +1 = x +1, if x ≥−1
−x −1, if x < −1
⎧⎨⎪
⎩⎪
x −1 = x −1, if x ≥1
1 −x, if x <1
⎧⎨⎪
⎩⎪
x +1 − x −1 dx
−2
2
∫ =?Problem
Solution
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x +1 − x −1 =
x +1 − x −1( ) , if x ≥1
x +1 − 1 −x( ) , if −1 ≤x <1
−x −1 − 1 −x( ) , if x < −1
⎧
⎨
⎪⎪
⎩
⎪⎪
x +1 − x −1 dx
−2
2
∫ =?Problem
Solution
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x +1 − x −1 =
2, if x ≥1
2 x , if −1 ≤x <1
2, if x < −1
⎧
⎨⎪⎪
⎩⎪⎪
x +1 − x −1 dx
−2
2
∫ =?Problem
Solution
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x +1 − x −1 dx
−2
2
∫ =?Problem
Solution
x +1 − x −1 =
2, if x ≥1
2 x, if 0 ≤x <1−2 x, if −1 ≤x < 0 2, if x < −1
⎧
⎨
⎪⎪
⎩
⎪⎪
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x +1 − x −1 dx
−2
2
∫ =?Problem
Solution
y = ||x + 1| - |x - 1||
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x +1 − x −1 dx
−2
2
∫ =?Problem
Solution
The integral is the area of
the yellow domain.
-2 21-1
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x +1 − x −1 dx
−2
2
∫ =?Problem
Solution
Area =
2 + 1 + 1 + 2
= 6.-2 21-1
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x +1 − x −1 dx
−2
2
∫ =?Problem
Answer
-2 21-1
x +1 − x −1 dx−2
2
∫ =6 .
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
1 2 3-3 -2 -1
2
3
1
Estimate using left Riemann sums
with 12 subintervals of equal length.
Problem
f x( )dx−3
3
∫ =?
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
f x( )dx−3
3
∫ =?
1 2 3-3 -2 -1
1
2
3Solution
Division points:
(-3,-2.5,-2,-1.5,-1,-0.5,0,0.5,1,
1.5,2,2.5,3).
As tag points tk, use the left end-points.
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
1 2 3-3 -2 -1
2
3
1
Problem
f x( )dx−3
3
∫ =?
tk f(tk)0.5 2.12
1 2.68
1.5 2.75
2 2.48
2.5 1.98
3 1.25
tk f(tk)-3 1.6
-2.5 1.76
-2 1.75
-1.5 1.37
-1 1.0
-0.5 1.0
0 1.5
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Answer
f x( )dx−3
3
∫ ≈ f tk( ) Δx =0.5 ⋅
k=1
12∑ f tk( )1
12∑tk f(tk)0.5 2.12
1 2.68
1.5 2.75
2 2.48
2.5 1.98
3 1.25
tk f(tk)-3 1.6
-2.5 1.76
-2 1.75
-1.5 1.37
-1 1.0
-0.5 1.0
0 1.5
Left(12) estimate = 0.5∙(1.6 + 1.76 + 1.75 + 1.37 + 1.0 + 1.0 + 1.5 + 2.12 + 2.68 + 2.75 + 2.48 + 1.98)
≈ 11
INTEGRALS FROM GRAPHS
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Average Value of a Function
1 2 3-3 -2 -1
2
3
1
Problem
f x( )dx−3
3
∫ ≈11
11
6≈1.8
The average value of the
function f on the interval [-3,3]
is ≈ 1.8.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Functions given by tables
Problem The speed of a racing car during the first 10 second of a race is given in the table below. Estimate the distance traveled during that time.
t (s) 0 1 2 3 4 5 6 7 8 9 10
s (m/s) 0 14 28 42 55 67 30 30 45 57 70
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem Estimate the distance traveled.
1234567
1 2 3 4 5 6 7 8 9 10
10 m/s
seconds
FUNCTIONS GIVEN BY TABLES
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem Estimate the distance traveled.
t (s) 0 1 2 3 4 5 6 7 8 9 10
s (m/s) 0 14 28 42 55 67 30 30 45 57 70
Time intervals: 1 second, Δt = 1 (s).
k 1 2 3 4 5 6 7 8 9 10
v 7 21 35 48.5 61 48.5 30 37.5 51 63.5
v = the average velocity during time interval.
FUNCTIONS GIVEN BY TABLES
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem Estimate the distance traveled.
Time intervals: 1 second, Δt = 1 (s).
k 1 2 3 4 5 6 7 8 9 10
v (m/s) 7 21 35 48.5 61 48.5 30 37.5 61 63.5
v = the average velocity during time interval.
d = distance traveled during time interval.
d (m) 7 21 35 48.5 61 48.5 30 37.5 51 63.5
FUNCTIONS GIVEN BY TABLES
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem Estimate the distance traveled.
Time
Speed
Distance traveled =
speed × time
= total area of the rectangles
FUNCTIONS GIVEN BY TABLES
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem Estimate the distance traveled.
k 1 2 3 4 5 6 7 8 9 10
v (m/s) 7 21 35 48.5 61 48.5 30 37.5 61 63.5
d (m) 7 21 35 48.5 61 48.5 30 37.5 51 63.5
Distance traveled during 10 seconds = 403 m.
Average speed 40.3 m/s ≈ 90 mph.
FUNCTIONS GIVEN BY TABLES
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem Estimate the distance traveled.
Time
Speed
s(t) = speed of an object at time t.
Distance traveled during time interval
[a,b]
= s t( )dt
a
b
∫ .
DISTANCE AS AN INTEGRAL OF SPEED
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Time
Speed
FORMULA 1 RACE CAR
Acceleration 0 to 200 km/h (124 mph): 3.8 s.
Deceleration: up to 5-6 g (48-58 m/s2).
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
x3 dx
0
10
∫ .
Approximate the value of the integral
Which method gives the best result?
COMPARING METHODS
Problem
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
x3 dx0
1
∫ .Approximate
COMPARING METHODS
Solution The integral is easy to compute:
x3 dx
0
1
∫ =x4
40
1
=14.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
COMPARING METHODS
Solution For this integral:
x3 dx0
1
∫ =14.
LEFT(1) = 0 MID(1) =1/8 RIGHT(1) = 1
TRAP(1) = 1/2
RIGHT(1) = 1
Problem
x3 dx0
1
∫ .Approximate
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
COMPARING METHODS
Solution For this integral:
x3 dx0
1
∫ =14.
RIGHT(1) = 1
SIMPSON(1)=
2 ⋅MID 1( ) + TRAP 1( )
3=
14+12
3=14.
Problem
x3 dx0
1
∫ .Approximate
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
COMPARING METHODS
Conclude
Simpson’s Approximation gives the precise value of the integral.
RIGHT(1) = 1
This is true for integrals of polynomials of degree at most 3.
Problem
x3 dx0
1
∫ .Approximate
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
e
−x2
2 dx0
2
∫ .
Approximate the integral
Estimate the errors.
INTEGRALS OF BELL SHAPED CURVES
-2 2
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
e−
x2
2 dx0
2
∫ ≈? Estimate the errors.
INTEGRALS OF BELL SHAPED CURVES
Solution
f x( ) =e
−x2
2
The function
is decreasing for 0 ≤ x ≤ 1.
Hence
RIGHT n( ) ≤ e
−x2
2 dx0
2
∫ ≤LEFT n( )
for all n.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
e−
x2
2 dx0
2
∫ ≈? Estimate the errors.
INTEGRALS OF BELL SHAPED CURVES
Solution Computing with a computer we get
RIGHT 10( ) ≈1 .109 ≤ e
−x2
2 dx0
2
∫ ≤LEFT 10( ) ≈1 .282 .
Error < 0.173.
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
e−
x2
2 dx0
2
∫ ≈? Estimate the errors.
INTEGRALS OF BELL SHAPED CURVES
Solution Observe that
f x( ) =e
−x2
2 ⇒ ′′f x( ) =x2 e−
x2
2 −e−
x2
2 .
⇒ ′′f x( ) > 0 for x >1,
and ′′f x( ) < 0 for −1 < x <1 .
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
e−
x2
2 dx0
2
∫ ≈? Estimate the errors.
INTEGRALS OF BELL SHAPED CURVES
Solution Hence the graph of f is concave down for -1 < x < 1, and concave up for x > 1 or x < -1.
TRAP n, 0,1⎡
⎣⎤⎦( ) +MID n, 1, 2⎡
⎣⎤⎦( ) ≤ e
−x2
2 dx0
2
∫
Hence
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
e−
x2
2 dx0
2
∫ ≈? Estimate the errors.
INTEGRALS OF BELL SHAPED CURVES
Solution Likewise
e
−x2
2 dx0
2
∫ ≤MID n, 0 ,1⎡⎣
⎤⎦( ) + TRAP n, 1, 2⎡
⎣⎤⎦( ).
This yields (with n = 10):
Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä
Problem
e−
x2
2 dx0
2
∫ ≈? Estimate the errors.
INTEGRALS OF BELL SHAPED CURVES
Solution We get e
−x2
2 dx0
2
∫ ≈1.1962 .
Computation with a computer algebra system, yields the more accurate estimate: