Block 5 Stochastic & Dynamic Systems Lesson 14 - Integral ...

Block 5 Stochastic & Dynamic Systems Lesson 14 - Integral ...

Block 5 Stochastic & Dynamic Systems Lesson 14 Integral Calculus The World is now a nonlinear, dynamic, and uncertain place. Block 5 - The End of the Road Lesson 14 - Integral Calculus Lesson 15 Stochastic (Probability) Models Lesson 16 Differential Equations Lesson 17 Dynamic Models f ( x)dx Integral Calculus on a silver platter Integration If F(x) is a function whose derivative F(x) = f(x), then F(x) is called the integral of f(x) For example, F(x) = x3 is an integral of f(x) = 3x2 Note also that G(x) = x3 + 5 and H(x) = x3 6 are also integrals of f(x) I like to call F(x) the antiderivative. Indefinite Integral

The indefinite integral of f(x), denoted by f ( x)dx F ( x) C where C is an arbitrary constant is the most general integral of f(x) The indefinite integral of f(x) = 3x2 is 2 3 3x dx x C A Strategy Gosh, it seems so simple. f ( x)dx F ( x) C the integrand First I guess at a function whose derivative is f(x) and then I add a constant of integration.

or use a table of integrals The top five n 1 x 1. x n dx C ; n 1 n 1 1 2. dx ln x C x 3. a dx ax C 4. e ax ax e dx C a n a bx 5. a bx dx

n 1 b n C Basic Rules of Integration 1. c f ( x)dx c f ( x)dx 2. f ( x) g ( x) dx f ( x) dx g ( x) dx 3. af ( x) bg ( x) dx a f ( x) dx b g ( x) dx The top four and the basic rules in action 10 2 .2 x 1.2 7 x 5e x 2 x 25 dx 1 2 .2 x 7 x dx 5e dx 10 dx 2x 1.2 dx 25dx x 3 .2 x .2 7 x 5e 2x

10 ln x 25 x C 3 .2 .2 Initial Conditions The rate at which annual income (y) changes with respect to years of education (x) is given by dy 100 x3/ 2 ; 4 x 16 dx where y = 28,720 when x = 9. Find y. y 100 x 3/ 2 dx 100 28, 720 40 9 5/ 2 x 3/ 2 1 3/ 2 1

C 40 x 5 / 2 C C 9720 C C 19, 000 y 40 x5/ 2 19, 000 Integrating au , a > 0 u a du ? let a eln a ln a u a du e u du e au e using e du C a au ln a u

ln a u e du C ln a Tricks of the Trade Its Magic Use some algebra 2 4 3 2 2 y y 2 y 2

3 y y dy y 3 3 dy 4 9 C 2 x 1 x 3 dx 1 3 2 x 5 x x 2 2 x 5 x 3 dx C 6 9 12 2 6

x 1 dx x x x 3 2 3 2 2 1 x 1 2 2 dx x x dx C x 2 x Adjusting for du method of substitution 1/ 2 1 2 x x 5 dx 2 2 x x 5 dx 2

1/ 2 1 1 x 5 2 x 5 2 xdx 2 2 3/ 2 2 u u du n 1 n 3/ 2 1 2 x 5 C 3 du u x 5 and 2 x dx 2 n 1 3/ 2

More dus 3 x2 3xe dx 2 e 2 xdx 2 du where u x ; 2 x dx 3 u 3 u e du e 2 2 3 x2 e C 2 x2 2x 1 x 2 5 dx x 2 5 2 xdx du 2 where u x 5; 2 x dx 1 2

du ln u ln x 5 C u Integration by Parts derived from the product rule for derivatives Slick Harry is trying to sell an ENM student a simple integration formula. u dv uv v du x xe dx ? x x

let u x; du dx and dv e dx; v e dx e x x x x x x xe dx xe e dx xe e e x 1 C x Another one?

u dv uv v du ln y dy ? dy let u ln y ; du and v y; dv dy y dy ln y dy ln y y y y y ln y y C y ln y 1 C Integration by Tables A favorite integration formula of engineering students is: dx x 1 px ln a be a be px a pa dx x 1 .1 x find :

ln 5 2 e C .1 x 5 2e 5 .1 5 There is no shame in using a Table of Integrals. Check it out! .1 x 2 e .1 d x 1

1 .1 x ln 5 2e C .1 x dx 5 .1 5 5 .5 5 2 e .5 5 2e .1x 5 .2e .1x .5 5 5 2e .1x 2.5 1 .1 x .1 x 5

2 e 2.5 5 2 e dx x 1 .1 x find : ln 5 2 e C .1 x 5 2e

5 .1 5 Another Table Problem find 7 x 2 ln 4 x dx n 1 n 1 u ln u u Table : u n ln u du C 2 n 1 n 1 set n 2, u 4 x, du 4dx 7 2 7 x ln 4 x dx 43 4 x ln 4 x 4dx 3 3 7 4 x ln 4 x 4 x 1 3 ln 4 x

C C 7 x 64 3 9 9 3 2 An Engineers Favorite Table ax e ax 1. xe dx 2 ax 1 a dx ex 2. x ln 1 e 1 ex 3. ln x dx x ln x x 2 2 x

x 4. x ln x dx ln x 2 4 no, this one The motivated student may wish to verify the above by showing that the derivatives of the right hand functions results in the corresponding integrands. The Definite Integral b f ( x)dx F (b) F (a) where F '( x) f ( x) a Areas under the curve Definite Integral Given a function f(x) that is continuous on the interval [a,b] we divide the interval into n subintervals of equal width, x, and from each interval choose a point, xi*. Then the definite integral of f(x) from a to b is Area under the curve f(x)

x x The Fundamental Theorem of Calculus Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be a function such that for all x in [a, b] then . Several engineering management students informally meeting after class to discuss the implications of the fundamental theorem Fundamental Theorem Evaluating a definite integral 2 2 4 x dx F (2) F (0) 0 4 23 3

32 0 3 3 4 x where F ( x) 4 x 2 dx 3 2 x 1 2 x 1 dx 0 33 1 26 3 3 3 3

3 2 0 2 1 3 3 0 1 3 3 The Area under a curve The area under the curve of a probability density function over its entire domain is always equal to one. Verify that the following function is a probability density function: 2 3t f (t ) 9 , 0 t 1000 10

1000 2 3 3t t dt 9 9 10 10 0 1000 0 3 3 10 10 9

1 Area between Curves Find the area bounded by y = 4 4x2 and y = x2 - 1 y1 = 4 4x2 y1 - y2 = 5 5x2 20 (5 5 x ) dx 1 3 1 (-1, 0) (1, 0) y 2 = x2 - 1 Improper Integrals r

f ( x) dx lim f ( x) dx a r a b b f ( x) dx lim f ( x) dx r 0 r f ( x) dx f ( x) dx f ( x) dx

0 and if the limit exists, the integral is said to be convergent. Otherwise it is divergent, right? Example an Improper Integral r 2 r 1 1 x dx lim 3 dx lim 3 r x

r x 2 1 1 1 1 1 1 lim 2 0 r 2r 2 2 2 1 Lets do another one r ke 0 x x

dx lim ke dx lim ke r 0 r lim ke k k r An engineering professor caught doing some improper integration. r x r 0 The Engineers Little Table of Improper Definite Integrals 1. e ax 0

1 dx a 2. xe ax 0 1 dx 2 a n ax 3. x e 0 n! dx n 1 a

4. x e dx 4 0 2 x2 A little table for engineers Some Applications Taking it to the limit The Crime Rate The total number of crimes is increasing at the rate of 8t + 10 where t = months from the start of the year. How many crimes will be committed during the last 6 months of the year? given : d crime dt 8t 10 12 8t 10 dt 4t

6 2 10t 12 6 4 144 120 4 36 60 492 Learning Curves Cumulative Cost Y (i ) ai b x hours to produce ith unit x T ( x) Y (i ) ai i 1 T ( x) V ( x) x

i 1 b cumulative direct labor hrs to produce x units average unit hours to produce x units Learning Curves Approximate Cumulative Cost x x b 1 ax T ( x) Y (i ) di a i di b 1 0 0 b b 1 b

ax ax V ( x) (b 1) x b 1 Learning Curves - example Production of the first 10 F-222s, the Air Forces new steam driven fighter, resulted in a 71 percent learning curve in dollar cost where the first aircraft cost $18 million. What will be cost of the second lot of 10 aircraft? (sim-lc 18e6 20 71) 20 .5059 20 T ( x) 18 i 10 .4941 18 x di .5059 10

18 .5059 .5059 20 10 $47.903 million .5059 ln(71/100) 47.903 b V ( x) $4.790 million ln 2 10 .4941 The Average of a Function The average or mean value of a function y = f(x) over the interval [a,b] is given by: b 1 y f ( x)dx

b a a Find the average of the function y = x2 over the interval [1,3]: 3 3 3 1 x 27 1 26 1 2 y x dx 4 3 1 1 6 6 3 3 2 1 Average profit An oil companys profit in dollars for the qth million gallons sold is given by P = P(q) = 369q 2.1q2 400 If the company sells 100 million gallons this year,

what is the average profit per gallon sold? 100 369q 2.1q 0 100 2 400 dq 2 3 100 1 369q 2.1q 400q 100 2 3 0

1 369(104 ) 2.1(106 ) 2 400(10 ) 100 2 3 104 369 210 0 4 3 100 2 $11,050 100 184.5 74 11, 050 or $.01105 6 10 An Inventory Problem

Demand for an item is constant over time at the rate of 720 per year. Whenever the on-hand inventory reaches zero, a shipment of 60 units is received. The inventory holding cost is based upon the average on-hand inventory. Let y = 60 720t be the on-hand inventory as a function of time where t is in years. It takes 60/720 = 1/12 yr to go from an inventory of 60 to 0. 1 y 1/12 0 1/12 1/12 60 720t 2 60 720t dt 12 60t 2 0 0 2

60 60 1 12 6 720 60 30 2 12 12 t Annuities Let A = the present value of a continuous annuity at an annual rate r (compounded continuously) for T years if a payment at time t is at the rate of f(t) per year. Then T rt A f (t )e dt 0 A is the present value of a continuous income stream Annuity Example Determine the present value of a continuous annuity at an annual rate of 8% for 10 years if the payment at

time t is at the rate of 1000t dollars per year. 10 A (1000t )e 10 .08 t 0 dt 1000 te .08 t dt 1000 0 e .08(10) 1 1000 .08(10) 1 2 2 .08 .08 1000 126.37377 156.25 $29,876 ax

e ax xe dx a 2 ax 1 e 10 .08t .08 2 .08t 1 0 Annuities Let S = the accumulated amount of a continuous annuity at an annual rate r (compounded continuously) for T years if a payment at time t is at the rate of f(t) per year. Then T S f (t )e 0

r (T t ) dt Back to the example 10 10 .08(10 t ) S 1000t e 0 .8 dt 1000e te .08t dt 0 10

e .08t 2225.541 .08t 1 2 .08 0 e .8 1 2225.541 .8 1 2 2 .08 .08 2225.541 126.3737 156.25 $66, 491 More of that darn example Recall continuous compounding S Pe rt .08 10 S $29,876e

$66, 490 Iterated Integrals Evaluate 1 1 0 0 2 3 3 y x dxdy 1 x2 0 0 ( x 2 xy y 2 )dydx

Double Integral Evaluate the integral over R where R is the triangle formed by y = x, y = 0, x = 1. 2 1 x 0 0 2 2 1 1 2 2 ( x y ) dA ( x y

) dydx ( x y )dxdy R 2 0 y

Recently Viewed Presentations

  • Le climat - KVHS M.J.Cyr

    Le climat - KVHS M.J.Cyr

    Climat La météorologie est la science qui étudie le climat Le climat est la description de la température sur une longue période de temps.
  • Chapter 4 Assessment and activities

    Chapter 4 Assessment and activities

    Chapter 4 Assessment and activities. Write the vocabulary word that best completes each sentence. Write a sentence for each word not used. Satrap. Agora. Democracy. ... They feared Persian conquest of Greece. Section 4—The Age of Pericles. 11. How was...
  • Proportions and Scale Drawings - Firelands Elementary School

    Proportions and Scale Drawings - Firelands Elementary School

    Proportions and Scale Drawings. Textbook Pages 430-435. Objective. I can solve scale drawing problems using proportions. Vocabulary. Scale Drawing: is the same shape but not the same size. A map is a scale drawing. The units are very important.
  • Adding a Preschool/Tuition-Based Care Session (Categories are ...

    Adding a Preschool/Tuition-Based Care Session (Categories are ...

    Adding a Care Session(Categories are the second level of your organization's offerings hierarchy: Program > Category > Class > Session). 4.) The Configure Recurrence box will open (the recurrence is the pattern that the session will occur).Select your Recurrence Criteria...
  • NNELS The National Network for Equitable Library Service (an ...

    NNELS The National Network for Equitable Library Service (an ...

    "National Network for Equitable Library Service ... "The Use and Abuse of Models of Disability" Disability & Society, Vol. 15, No. 1, 2000, pp. 157 - 165. 1. Medical Model. Disability is the result of physiological impairment due to injury...
  • The Problem of Evil - Beechen Cliff School Humanities Faculty

    The Problem of Evil - Beechen Cliff School Humanities Faculty

    No 'problem of evil' for atheists. In a universe of electrons and selfish genes, blind physical forces and genetic replication, some people are going to get hurt, other people are going to get lucky, and you won't find any rhyme...
  • Orbis Pictus: Picturing the World and Imprinting the

    Orbis Pictus: Picturing the World and Imprinting the

    Comenius' didactic principles have to be read in the context of early Seventeenth Century assumptions about the world, human nature, and the purpose of life. He did not live in our open, evolving world, but the divinely crafted cosmos pictured...
  • Day 6: Invasive Species - AP Environmental Science

    Day 6: Invasive Species - AP Environmental Science

    Non-native species. H. ave no natural enemies so they can thrive in the new ecosystem and crowd out the native species: Examples. Keystone species: Their strong interactions with other species affect the health and survival of these species.