5.6 Substitution Method
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1 5.6 Substitution Method Recll the Chin Rule: (f(g(x))) = f (g(x))g (x) Wht hppens if we wnt to find f (g(x))g (x) dx? The Substitution Method: If F (x) = f(x), then f(u(x))u (x) dx = F (u(x)) + C. Steps: Exmples: 2x cos(x 2 ) dx sin θ cos 5 θ dθ 1
2 1 x ln x dx x 2 e x3 dx Chnge of Vribles Formul for Definite Integrls: If u(x) is differentible on [, b] nd f(x) is integrble on the rnge of u(x), then b f(u(x))u (x) dx = u(b) u() f(u) du e z + 1 e z + z dz x 2 (1 + 2x 3 ) 5 dx 2
3 6.1 Are Between Two Curves Are between grphs= Steps: b (y top y bot )dx = b (f(x) g(x))dx Exmples: Sketch the region enclosed by the given curves. Then, find the re of the region. y = x + 1, y = 9 x 2, x = 1, x = 2 y = x, y = x 2 3
4 x = 2y 2, x = 4 + y 2 Exmple: y = 8 x 2, y = 8x, y = x 4
5 6.2 Setting Up Integrls: Volume, Averge Vlue Volume s the Integrl of Cross-Sectionl Are: Suppose tht solid body extends from height y = to y = b. Let A(y) be the re of the horizontl cross section t height y. Then Volume of the solid body= b A(y)dy Find the volume of the solid S: the bse of S is the prbolic region {(x, y) x 2 y 1}. Cross-sections perpendiculr to the y-xis re equilterl tringles. Find the volume of right circulr cone with height h nd rdius r. Averge Vlue How do we clculte the verge of n numbers? 5
6 Wht if we wnt to clculte the verge vlue of function on prticulr intervl? Averge Vlue: The verge vlue of n integrble function f(x) on [, b] is the quntity f ve = 1 b b f(x) dx. Ex: () Find the verge vlue of f(x) = ln x on the intervl [1, 3]. 6
7 Men Vlue Theorem for Integrls: If f(x) is continuous on [, b], then there exists vlue c [, b] such tht f(c) = f ve = 1 b f(x) dx. b (b) Find c such tht f ve = f(c). 7
8 6.3 Volumes of Revolution Volume of Solid of Revolution: Disc Method: If f(x) is continuous nd f(x) 0 on [, b], the the volume V obtined by rotting the region under the grph bout the x-xis is [with R = f(x): V = π b R 2 dx = π b f(x) 2 dx. Ex: Find the volume of the solid obtined by rotting the region bounded by the given curves round the specified line. y = 1 x 2, y = 0; bout the x-xis. Volume of Solid of Revolution: Wsher Method: Let R = f(x) be the outer rdius nd r = g(x) be the inner rdius, then V = π b EX: y = x 2/3, x = 1, y = 0; bout the y-xis (R 2 r 2 ) dx = π b (f(x) 2 g(x) 2 ) dx 8
9 EX: y = 1/x, y = 0, x = 1, x = 3; bout y = 1 9
10 6.4 The Method of Cylindricl Shells Volume of Solid of Revolution: Shell Method: The volume V of the solid obtined by rotting the region under the grph y = f(x) over the intervl [, b] bout the y xis is equl to V = 2π Note: b xf(x) dx = 2π b (rdius)(height of shell) dx. EX: Let S be the solid obtined by rotting bout the y-xis the region bounded by y = x(x 1) 2 nd y = 0. Explin why cylindricl shells is the best method to use. EX: Find the volume of the solid obtined by rotting the region bounded by y = x x 2 nd y = 0 bout the line x = 2. 10
11 5.6 Integrtion by Prts Recll the Product Rule: (u(x)v(x)) = Integrtion By Prts Formul: u(x)v (x) dx = u(x)v(x) u (x)v(x) dx How to choose u nd v : u: v: Exmples: 4. xe x dx ln x dx 11
12 x 2 cos x dx e θ cos 2θ dθ π/2 0 x sin x dx (2xe x2 ) sin(ln e x2 ) dx Reduction formuls Prove the reduction formul: x n cos x dx = x n sin x n x n 1 sin x dx. 12
13 5.7 Techniques of Integrtion: Trigonometric Integrls Recll your trig identities!!!! Exmples: Odd power of sin x: sin 3 x dx Odd power of sin x or cos x: sin 4 x cos 5 x dx 13
14 Even powers of sin x nd cos x: sin 2 x cos 4 x dx 8. π/4 0 tn 2 x sec 4 x dx 14
15 5.7 Techniques of Integrtion: Trigonometric Substitution Tringles! Steps: Deciding on substitution: Squre Root Form in Integrnd Trigonometric Substitution 2 x x 2 x 2 2 Exmples: 10. x 2 1 x 4 dx 15
16 x 3 16 x 2 dx 32. x 3 2x x 2 dx 16
17 5.7 Techniques of Integrtion: Prtil Frctions Prtil Frction Decomposition: Liner Fctors vs. Qudrtic Fctors: Liner: Repeted Liner: Qudrtic: Repeted Qudrtic: Exmple: Give the form of the prtil frction decomposition: 2x + 4 (x 2)(x 2 + 4) Solving for the constnts: Solve for the constnts in the previous problem: 17
18 Integrting x 1 x 2 + 3x + 2 dx 22. 2x (x 2 + 1)(x 2 + 4) dx Long Division 28. x 3 4x 10 x 2 x 6 dx 18
19 5.9 Approximte Integrtion Recll: Riemnn Sums Midpoint Rule: The N th midpoint pproximtion to b f(x) dx is M N = x(f(c 1 ) + f(c 2 ) + + f(c N ) where x = b N nd c j is the midpoint of the jth intervl [x j 1, x j ] or c j = +(j.5) x. Trpezoidl Rule: The N th trpezoidl pproximtion to b T N = 1 2 x(y 0 + 2y y N 1 + y N ) f(x) dx is where x = b N nd y j = f( + j x). 19
20 Simpson s Rule: Assume tht N is even. Let x = b N nd y j = f( + j x). The Nth pproximtion to b f(x) dx by Simpson s Rule is S N = 1 3 x[y 0 + 4y 1 + 2y y N 3 + 2y N 2 + 4y N 1 + y N ]. Exmple: 8. Use () the Trpezoidl Rule, (b) the Midpoint Rule, nd (c) Simpson s Rule to pproximte 1/2 0 sin(x 2 ) dx, with n = 4. 20
21 Error Bounds: How ccurte re these estimtes? Error Bound for T N nd M N : Let K 2 be ny number such tht f (x) K 2 for ll x [, b]. Then E T =Error(T N ) K 2(b ) 3 12N 2, E N =Error(M N ) K 2(b ) 3 24N 2 Error Bound for S N : Let K 4 be the number such tht f (4) K 4 for ll x [, b]. Then E S =Error(S N ) K 4(b ) 5 180N 4 e x dx. Clculte the correspond- 19. () Assume T 10 = nd S 10 = for ing errors E T nd E S using the formuls given bove. 1 0 (c) How lrge do we hve to choose N so tht the pproximtions T N nd S N to the integrl in prt () re ccurte to within ? 21
22 5.10 Improper Integrls Two min types of improper integrls: Infinite Intervls Three cses: Convergent vs. Divergent: Exmples: Determine whether ech of the following is convergent or divergent. Evlute those tht re convergent dx 6. (3x + 1) x 5 dx 12. (2 v 4 ) dv 22
23 1 Exmple: For wht vlues of p does the integrl dx converge? 1 xp ** Discontinuous Integrnds Three cses: dx 24. x x dx 23
24 (x 1) 1/5 dx Comprison Test: Assume tht f(x) g(x) 0 for x. If If f(x) dx converges, then g(x) dx diverges, then g(x) dx converges. f(x) dx diverges. Exmples: Use the Comprison Test to determine whether the integrl is convergent or divergent e x x dx x 1 + x 6 dx 24
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