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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: