Integration. 5.1 Antiderivatives and Indefinite Integration. Suppose that f(x) = 5x 4. Can we find a function F (x) whose derivative is f(x)?

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1 5 Integration 5. Antiderivatives and Indefinite Integration Suppose that f() = 5 4. Can we find a function F () whose derivative is f()? Definition. A function F is an antiderivative of f on an interval I if F () =f() for all in I. Note. Observe that F () = 5, F 2 () = 5 + 2, and F 3 () = 5 7 are all antiderivatives of f. To represent all possible antiderivatives of f(), we write F () = 5 + C where C is a constant. Theorem 5.. If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and onl if G is of the form G() =F ()+C, for all in I where C is a constant. Note. We call C the constant of integration and G the general antiderivative of f. Notation. From the last section, for we wrote d d = f() d = f()d The process of finding all solutions to this equation is called indefinite integration or antidifferentiation. We write =

2 2 5 Integration Note. There is an inverse relationship between differentiation and indefinite integration. That is F ()d = If f()d = F ()+C, then [ d d ] f()d = Basic Integration Rules Compare these with the basic differentiation rules that we alread know.. 0d = 2. k d = 3. kf()d = [f() ] ± g() d = n d = d = cos d = sin d = sec 2 d = csc 2 d = sec tan d = csc cot d =

3 3 Eample. Evaluate 5d 5. Antiderivatives and Indefinite Integration Eample 2. Evaluate 5 d Eample 3. Evaluate 4 d Eample 4. Evaluate 3 d Eample 5. Evaluate 2 cos d Eample 6. Evaluate 5 d

4 4 5 Integration Eample 7. Evaluate d Eample 8. Evaluate ( + 4) d Eample 9. Evaluate ( ) d Eample 0. Evaluate 3 +2 d Eample. Evaluate sin cos 2 d

5 5 5. Antiderivatives and Indefinite Integration Eample 2. Near the surface of the earth, the acceleration of a falling bod due to gravit is 32 feet per second per second, provided that air resistance is neglected. If an object is thrown upward from an initial height of 000 feet with an initial velocit of 50 feet per second, find its velocit and height four seconds later.

6 6 5 Integration 5.2 Area Sigma Notation The sum of n terms a, a 2, a 3,..., a n ma be denoted b n a i = i= where i is the inde of summation, a i is the i th term (or general term) of the sum, and and n are the lower and upper bounds of summation respectivel. Eample. Epand 5 i. i= Eample 2. Epand 6 i 2. i=3 Eample 3. Epand n k= n (k2 ). Eample 4. Epand n f( i ). i=

7 7 5.2 Area Properties of Sigma Notation n. ka i = i= n 2. (a i ± b i )= i= Theorem n c = 2. i= n i 2 = 4. i= n i = i= n i 3 = i= Eample 5. Find the summation formula for n = 0, 00, 000, n i= i 2 +. Use our result to evaluate the sum for n 3

8 8 5 Integration Eample 6. Use four rectangles to find two approimations of the area of the region ling between the graph of f() =2 and the -ais between the the vertical lines = 0 and =

9 9 5.2 Area To approimate the area under the curve of a nonnegative function bounded b the -ais and the line = a and = b...

10 0 5 Integration

11 5.2 Area Theorem 5.3. Let f be continuous and nonnegative on the interval [a, b]. The limits of both the lower and upper sums eist and lim s(n) = n where =(b a)/n and f(m i ) and f(m i ) are the minimum and maimum values of f on the subinterval. Definition. Let f be continuous and nonnegative on the interval [a, b]. The area of the region bounded b the graph of f, the -ais, and the vertical lines = a and = b is area = where =(b a)/n. Eample 7. Find the area of the region bounded b the graph of f() = 2, the ais, and the vertical lines = 0 and =.

12 2 5 Integration 5.3 The Riemann Sum and the Definite Integral The following definition allows us to find the area under a function when the width of the rectangles are not all the same. Definition. Let f be defined on the closed interval [a, b], and let be a partition of [a, b] given b a = where i is the width of the i th subinterval. If c i is an point in the i th subinterval, then the sum n f(c i ) i i= is called a Riemann sum of f for the partition. Note. This is a generalization of the upper and lower sum method we developed earlier. We define the width of the largest subinterval of a partition to be the norm of the partition denoted b. If ever subinterval is of equal width, then the partition is said to be regular and the norm is given b =

13 3 5.3 The Riemann Sum and the Definite Integral If we are using a general partition, then the norm of the general partition is related to the number of subintervals of [a, b] b b a and as 0, then this implies that n. The converse of this statement is not true in general. It is onl true in the case of a regular partition. For instance, consider the partition n =2 n of the interval [0, ]. Thus, lim 0 n f(c i ) i = L i= eists provided that for each ε> 0, there eists a δ> 0 with the propert that for ever partition with <δ implies that n L f(c i ) i <ε for an c i in the i th subinterval of. Definition. If f is defined on the closed interval [a, b] and the limit lim 0 n f(c i ) i i= eists (as described above), then f is integrable on [a, b] and the limit is denoted b lim 0 n f(c i ) i = i= i= and 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.

14 4 5 Integration Theorem 5.4. If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b]. Eample. Evaluate 2 2 d

15 5 5.3 The Riemann Sum and the Definite Integral Theorem 5.5. If a function f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded b the graph of f, the -ais, and the vertical lines = a and = b is given b area = Eample 2. Sketch the region corresponding to geometr of the region. 5 3d and evaluate the integral b using the Eample 3. Sketch the region corresponding to the geometr of the region d and evaluate the integral b using

16 6 5 Integration Theorem 5.6. If a function f is integrable on the three closed intervals determined b a < c < b, then b a f()d = Eample 4. Evaluate 4 d Theorem 5.7. Suppose that k is a real number and f and g are integrable on [a, b]. Then, the function kf and f ± g are integrable on [a, b] and. 2. b a b a kf()d = [ f() ± g() ] d = Eample 5. For the definite integral, we can write 3 ( ) d =

17 7 5.3 The Riemann Sum and the Definite Integral Theorem If f is integrable and nonnegative on the closed interval [a, b], then 2. If f and g are integrable on the closed interval [a, b] and f() g() for ever in [a, b], then Properties of the Definite Integral. If f is defined at = a, then 2. If f is integrable on [a, b], then Eample 6. For the given definite integrals, we can write π π and 0 3 e d = ( + 4) d =

18 5 Integration 5.4 The Fundamental Theorem of Calculus Theorem 5.9 (Fundamental Theorem of Calculus Part I). If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on [a, b], then b a f()d = Note. We lose the constant of integration when performing definite integration. 8

19 9 5.4 The Fundamental Theorem of Calculus Eample. Evaluate 2 ( ) d. Eample 2. Evaluate 4 0 d. Eample 3. Evaluate π 0 tan d. Eample 4. Find the area of the region bounded b the graph of =, the -ais, and the vertical lines = and =e 2.

20 5 Integration Theorem 5.0 (The Mean Value Theorem for Integrals). If f is continuous on the closed interval [a, b], then there eists a number c in the closed interval [a, b] such that b a f()d = Eample 5. Use the Mean Value Theorem for Integrals to find all values of c with that satisf the theorem. 0 (3 2 2)d 20

21 2 5.4 The Fundamental Theorem of Calculus Definition. If f is integrable on the closed interval [a, b], then the average value of f on the interval is given b average value = Eample 6. Find the average value of f() = 2 on the interval [, 4].

22 22 5 Integration Accumulation Functions Define the accumulation function F () = where F is a function of, f is a function of t, and a is a constant. Eample 7. Evaluate F () = ( t)dt for =0, 4, 2, 3 4,. 0 t t t t t

23 The Fundamental Theorem of Calculus Theorem 5. (Fundamental Theorem of Calculus Part II). If f is continuous on an open interval I containing a, then, for ever in the interval, [ d ] f(t)dt = d a

24 24 5 Integration Eample 8. Find the derivative of 0 t2 dt. 3 Eample 9. Find the derivative of sin t dt. 0

25 The u-substitution 5.5 The u-substitution We want to now be able to work with integrands that have the form of the Chain Rule. Hence, recall that Theorem 5.2 (The u-substitution). Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then f ( g() ) g ()d = If u = g(), then du = g ()d and f(u)du = Eample. Evaluate ( 2 + 4)(2)d. Eample 2. Evaluate 2e 2 d. Eample 3. Evaluate ( 2 + 4) 2 d.

26 26 5 Integration 3 Eample 4. Evaluate d. Eample 5. Evaluate 3 d. Eample 6. Evaluate sin(5) cos(5) d.

27 The u-substitution Theorem 5.3 (The General Power Rule for Integration). If g is a differentiable function of, then [g() ] n g ()d = If u = g(), then u n du = Eample 7. Evaluate 2 ( 2 ) 2 d. Eample 8. Evaluate sin 2 cos d.

28 5 Integration Theorem 5.4 (Change of Variables for Definite Integrals). If the function u = g() has a continuous derivative on the closed interval [a, b] and f is continuous on the range of g, then b a f ( g() ) g ()d = Eample 9. Evaluate 0 ( 2 ) 3 d. Eample 0. Evaluate d. 28

29 The u-substitution Theorem 5.5. Let f be integrable on the closed interval [ a, a].. If f is an even function, then a a f()d = 2. If f is an odd function, then a a f()d = Eample. Evaluate 0 0 ( 3 + )d. Eample 2. Evaluate 2 2 (4 2 )d.

30 5 Integration 5.6 Numerical Integration Theorem 5.6 (The Trapezoid Rule). Let f be continuous on [a, b]. Then, b f()d a Eample. Use the Trapezoid Rule to approimate 2 d for n = 5. 30

31 5.6 Numerical Integration Theorem 5.7. Let p() =A 2 + B + C, then b p()d = a Theorem 5.8 (Simpson s Rule). Let f be continuous on [a, b]. If n is even, then b a f()d Eample 2. Use Simpson s Rule to approimate 2 d for n = 0. 3

32 32 5 Integration Theorem If f has a continuous second derivative on [a, b], then the error of estimating the Trapezoid Rule is b a f()d b E 2. If f has a continuous fourth derivative on [a, b], then the error of estimating Simpson s Rule is b a f()d b E Eample 3. Determine a value of n such that the Trapezoid Rule will approimate error that is less than d with an

33 Integration and the Natural Logarithm Using what we know about the derivative of the natural logarithm, 5.7 Integration and the Natural Logarithm Theorem Let u be a differentiable function of. d. = du 2. u = u d 3. u = Eample. Evaluate 3 d. Eample 2. Evaluate 5 2 d.

34 34 5 Integration Eample 3. Find the area of the region bounded b the graph of = vertical lines = 0 and =e 2. 2, the -ais, and the Eample 4. Evaluate d. 2 +

35 35 Eample 5. Solve the differential equation d d = e ln. 5.7 Integration and the Natural Logarithm Eample 6. Evaluate e ( + ) 2 d.

36 36 5 Integration Eample 7. Evaluate cot d. Eample 8. Evaluate csc d.

37 Integrating Inverse Trigonometric Functions Theorem 5.2. Let u be a differentiable function of, and let a>0. du. a2 u = du a 2 + u 2 = du u u 2 a 2 = 5.8 Integrating Inverse Trigonometric Functions

38 5 Integration Eample. Evaluate d 49 2 Eample 2. Evaluate d 3+e 2 Eample 3. Evaluate d

39 Integrating Inverse Trigonometric Functions Eample 4. Evaluate d 4 9 Eample 5. Evaluate d

40 40 5 Integration Completing the Square Eample 6. Evaluate 2 2 d Eample 7. Evaluate d