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Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 5 Integration
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Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.1 Estimating with Finite Sums (1 st lecture of week 03/09/07- 08/09/07)
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Slide 5 - 3 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Riemann Sums Approximating area bounded by the graph between [a,b]
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Slide 5 - 4 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Partition of [a,b] is the set of P = {x 0, x 1, x 2, … x n-1, x n } a = x 0 < x 1 < x 2 …< x n-1 < x n =b c n [x n-1, x n ] ||P|| = norm of P = the largest of all subinterval width Area is approximately given by f(c 1 ) x 1 + f(c 2 ) x 2 + f(c 3 ) x 3 + … + f(c n ) x n
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Slide 5 - 5 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Riemann sum for f on [a,b] R n = f(c 1 ) x 1 + f(c 2 ) x 2 + f(c 3 ) x 3 + … +f(c n ) x n
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Slide 5 - 6 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Let the true value of the area is R Two approximations to R: c n = x n corresponds to case (a). This under estimates the true value of the area R if n is finite. c n = x n-1 corresponds to case (b). This over estimates the true value of the area S if n is finite. go back Figure 5.4
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Slide 5 - 7 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Limits of finite sums Example 5 The limit of finite approximation to an area Find the limiting value of lower sum approximation to the area of the region R below the graphs f(x) = 1 - x 2 on the interval [0,1] based on Figure 5.4(a)Figure 5.4(a)
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Slide 5 - 8 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution x k = (1 - 0)/n= 1/n ≡ x; k = 1,2,…n Partition on the x-axis: [0,1/n], [1/n, 2/n],…, [(n-1)/n,1]. c k = x k = k x = k/n The sum of the stripes is R n = x 1 f(c 1 ) + x 2 f(c 2 ) + x 3 f(c 3 ) + …+ x n f(c n ) x f(1/n) + x f(2/n) + x f(3/n) + …+ x n f(1) = ∑ k=1 n x f(k x) = x ∑ k=1 n f (k/n) = (1/ n) ∑ k=1 n [1 - (k/n = ∑ k=1 n 1/ n - k /n = 1 – (∑ k=1 n k / n = 1 – [ n n+1 n+1 ]/ n = 1 – [2 n n +n]/(6n ∑ k=1 n k n n+1 n+1
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Slide 5 - 9 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Taking the limit of n → ∞ The same limit is also obtained if c n = x n-1 is chosen instead. For all choice of c n [x n-1,x n ] and partition of P, the same limit for S is obtained when n ∞
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Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.3 The Definite Integral (2 nd lecture of week 03/09/07- 08/09/07)
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Slide 5 - 11 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 12 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley “The integral from a to b of f of x with respect to x”
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Slide 5 - 13 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The limit of the Riemann sums of f on [a,b] converge to the finite integral I We say f is integrable over [a,b] Can also write the definite integral as The variable of integration is what we call a ‘dummy variable’
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Slide 5 - 14 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Question: is a non continuous function integrable?
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Slide 5 - 15 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Integral and nonintegrable functions Example 1 A nonintegrable function on [0,1] Not integrable
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Slide 5 - 16 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Properties of definite integrals
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Slide 5 - 17 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 18 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 19 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 Finding bounds for an integral Show that the value of is less than 3/2 Solution Use rule 6 Max-Min Inequality
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Slide 5 - 20 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Area under the graphs of a nonnegative function
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Slide 5 - 21 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 Area under the line y = x Compute (the Riemann sum) and find the area A under y = x over the interval [0,b], b>0
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Slide 5 - 22 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution By geometrical consideration: A=(1/2) high width= (1/2) b b= b 2 /2
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Slide 5 - 23 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Using geometry, the area is the area of a trapezium A= (1/2)(b-a)(b+a) = b 2 /2 - a 2 /2 Using the additivity rule for definite integration: Both approaches to evaluate the area agree
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Slide 5 - 24 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley One can prove the following Riemannian sum of the functions f(x)=c and f(x)= x 2:
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Slide 5 - 25 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Average value of a continuous function revisited Average value of nonnegative continuous function f over an interval [a,b] is In the limit of n ∞, the average =
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Slide 5 - 26 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 27 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 28 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 Finding average value Find the average value of over [-2,2]
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Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.4 The Fundamental Theorem of Calculus (2 nd lecture of week 03/09/07-08/09/07)
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Slide 5 - 30 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Mean value theorem for definite integrals
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Slide 5 - 31 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 32 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 33 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 Applying the mean value theorem for integrals Find the average value of f(x)=4-x on [0,3] and where f actually takes on this value as some point in the given domain. Solution Average = 5/2 Happens at x=3/2
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Slide 5 - 34 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fundamental theorem Part 1 Define a function F(x): x,a I, an interval over which f(t) > 0 is integrable. The function F(x) is the area under the graph of f(t) over [a,x], x > a ≥ 0
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Slide 5 - 35 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 36 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fundamental theorem Part 1 (cont.) The above result holds true even if f is not positive definite over [a,b]
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Slide 5 - 37 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Note: Convince yourself that (i)F(x) is an antiderivative of f(x)? (ii)f(x) is an derivative of F(x)?
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Slide 5 - 38 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley F(x) is an antiderivative of f(x) because F'(x)= f(x) F(x)F(x)f(x) = F'(x) d/dx f(x) is an derivative of F(x)
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Slide 5 - 39 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 Applying the fundamental theorem Use the fundamental theorem to find
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Slide 5 - 40 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 Constructing a function with a given derivative and value Find a function y = f(x) on the domain (- /2, /2) with derivative dy/dx = tan x that satisfy f(3)=5. Solution Set the constant a = 3, and then add to k(3) = 0 a value of 5, that would make k(3) + 5 = 5 Hence the function that will do the job is
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Slide 5 - 41 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fundamental theorem, part 2 (The evaluation theorem)
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Slide 5 - 42 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To calculate the definite integral of f over [a,b], do the following 1. Find an antiderivative F of f, and 2. Calculate the number
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Slide 5 - 43 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To summarise
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Slide 5 - 44 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 Evaluating integrals
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Slide 5 - 45 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7 Canceling areas Compute (a) the definite integral of f(x) over [0,2 ] (b) the area between the graph of f(x) and the x-axis over [0,2 ]
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Slide 5 - 46 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 8 Finding area using antiderivative Find the area of the region between the x- axis and the graph of f(x) = x 3 - x 2 – 2x, -1 ≤ x ≤ 2. Solution First find the zeros of f. f(x) = x(x+1) (x-2)
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Slide 5 - 47 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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5.5 Indefinite Integrals and the Substitution Rule (3 rd lecture of week 03/09/07-08/09/07)
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Slide 5 - 49 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Note The indefinite integral of f with respect to x is a function plus an arbitrary constant A definite integral is a number.
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Slide 5 - 50 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The power rule in integral form From we obtain the following rule
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Slide 5 - 51 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 Using the power rule
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Slide 5 - 52 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2 Adjusting the integrand by a constant
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Slide 5 - 53 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Substitution: Running the chain rule backwards Used to find the integration with the integrand in the form of the product of
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Slide 5 - 54 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 Using substitution
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Slide 5 - 55 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 Using substitution
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Slide 5 - 56 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 Using Identities and substitution
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Slide 5 - 57 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 Using different substitutions
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Slide 5 - 58 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The integrals of sin 2 x and cos 2 x Example 7
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Slide 5 - 59 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The integrals of sin 2 x and cos 2 x Example 7(b)
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Slide 5 - 60 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 8 Area beneath the curve y=sin x 2 For Figure 5.24, find (a) the definite integral of g(x) over [0,2 ]. (b) the area between the graph and the x- axis over [0,2 ].
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Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.6 Substitution and Area Between Curves (3 rd lecture of week 03/09/07-08/09/07)
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Slide 5 - 62 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Substitution formula
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Slide 5 - 63 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 Substitution Evaluate
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Slide 5 - 64 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2 Using the substitution formula
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Slide 5 - 65 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definite integrals of symmetric functions
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Slide 5 - 66 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 67 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 Integral of an even function How about integration of the same function from x=-1 to x=2
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Slide 5 - 68 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Area between curves
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Slide 5 - 69 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 70 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 71 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the area of the region enclosed by the parabola y = 2 – x 2 and the line y = -x. Example 4 Area between intersecting curves
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Slide 5 - 72 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the area of the shaded region Example 5 Changing the integral to match a boundary change
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Slide 5 - 73 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 5 - 74 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 Find the area of the region in Example 5 by integrating with respect to y
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