Finl Review, Mth 860 Thoms Clculus Erly Trnscendentls, 2 ed 6. Applictions of Integrtion: 5.6 (Review Section 5.6) Are between curves y = f(x) nd y = g(x), x b is f(x) g(x) dx nd similrly for x = f(y) nd x = g(y). b 6. Volumes by slices: b A(x)dx where A(x) is the re of the cross section perpendiculr to the x-xis. Specil cse is the solid of rottion obtined by rotting the region between curves y = f(x) nd y = g(x), x b bout the x-xis: b π f(x) 2 g(x) 2 dx. Wht hppens in the cse the xis of rottion is y = 3? There is n nlogous formul obtined by interchnging x nd y. Exmple: The region between the curves x = y 2 nd y = x 2 is rotted bout the y-xis. Find the volume of the solid generted. 6.2 The method of cylindricl shells: the region between curves y = f(x) nd y = g(x), x b bout the y-xis: b 2πx[f(x) g(x)]dx. Wht hppens when the xis of rottion is x = 2? Agin interchnging x nd y gives n nlogous formul. Exmple: Rotte the region in the previous exmple round the x-xis. 6.3 Arc Length. The curve y = f(x), x b hs length b +(f (x)) 2 dx 6.4 Are of Surfces of Rottion: The curve y = f(x), x b is totted bout the x xis. Find the re of the surfce. 2π b 7. Trnscendentl Functions: f(x) +(f (x)) 2 dx 7. The Nturl Logrithm lnx = x (/t)dt hs hs the property ln(xy) = lnx+ lny etc. (d/dx)lnu = (/u)du/dx. Wht is tnxdx? 7. e x, the Nturl Exponentil is the inverse of lnx: e lnx = x = ln(e x ). (d/dx)e u = e u du/dx nd we sw exp(x) = e x if x is rtionl nd so for ll 7. Generl logs nd exponentils. Differentite y = 3 secx. or y = log 2 x. 7.3 Hyperbolic Functions: coshx = ex +e x ; sinhx = ex e x 2 2 8. Techniques of Integrtion: nd tnhx = sinhx coshx.
2 8. Integrtion by prts: udv = uv vdu. On p 407, wht is u nd dv in questions -24? 8.2 Integrtion of powers of Trig functions Identities: () (sinx) 2 +(cosx) 2 = (b) (cosx) 2 = 2 (+cos2x) (c) (sinx) 2 = 2 ( cos2x) (d) (secx) 2 = (tnx) 2 + (e) (cscx) 2 = (cotx) 2 + (f) sin2x = 2sinxcosx (g) cos2x = (cosx) 2 (sinx) 2 (sinx) m (cosx) n dx Cses: m is odd (u = cosx); n is odd (u = sinx); m nd n even (cosx) 2 = 2 (+cos2x) (sinx)2 = 2 ( cos2x) 8.2 Integrtion of powers of Trig functions (tnx) m (secx) n dx If n is even substitute u = tnx. If m is odd then substitute u = secx. Also secxdx = ln secx+tnx +C Also 8.3 Trig Substitution secxdx = ln secx+tnx +C For Integrls Involving Substitute Use the Identity 2 x 2 x = sinθ, dx = cosθdθ 2 x 2 = cosθ 2 +x 2 x = tnθ, dx = (secθ) 2 dθ 2 +x 2 = secθ 8.4 Prtil Frctions.. Divide; 2. Fctor divisor 3. Expnd by prtil frctions 4. Solve for coefficients 5. Integrte.
3 8 Evlute xrctnxdx = (cosx)(sinx) 6 dx = (cosx) 4 dx = (tnx) 3 dx = v 2 v 2 dv = +x 2 (+x) 3 dx 8.7 Improper integrls. Type I nd II. dx < if nd only if p >. xp. Comprison Test. Assume 0 f(x) g(x). Then g(x)dx < implies f(x)dx < OR f(x)dx = implies g(x)dx = 0. Series: 0. Sequences lnn () lim n = 0 (b) lim n /n = (c) lim x /n = (d) lim x n = 0 if x <. (e) lim (+ x n x n (f) lim n! = 0 2. Series. ) n = e x 0.2 Geometric series +r+r 2 +r 3 = /( r). 0.2 Telescoping series. 0.2 nth term test for divergence. lim n n 0 0.3 Integrl Test 0.3 p-series n /np converges iff p >. 0.4 Comprison Test n /(n2 +n)
4 0.4 Limit Comprison Test. n /(n2 n) 0.6 Conditionl or bsolute convergence? 0.5 Alternting Series Test. Is n ( )n / n conditionlly convergent? AST is only for conditionl convergence. 0.6 Rtio Test. lim n n+ / n 0.6 Root Test. lim n n /n 0.7 Power series c n(x ) n. Rdius of convergence nd intervl of convergence. Rtio or root test. Check end points? 0.8 Integrtion nd differentition of power series: f(x) = c n(x ) n f (x) = nc n (x ) n n= f(x)dx = C + c n n+ (x )n+ All three power series hve the sme rdius of convergence. 0.8 The geometric series nd power series. For exmple, /x = /( + (x )) = (x )+(x ) 2 (x ) 3 (Here r = (x ) nd =. Convergence if r = x <. 0.9 Tylor series nd Mclurin series ( = 0) f (n) () (x ) n n! nd Tylor polynomils, T N (x) of degree N is P N (x) = N f (n) () (x ) n n! 0.9 Common Mclurin series: e x = +x+ x2 2! xn n! cosx = x2 2! + x4 4! + ( )n x 2n (2n)! sinx = x x3 3! + x5 5! + ( )n x 2n+ (2n+)! 0.0 Integrtion of power series like tht of e x2 Prmetric nd Polr Curves.. Prmetric Curves. Grphing
5.2 Prmetric Curves nd Clculus. dy dx = dy/dt. Tngent line to prmetric curve. dx/dt b Length (compre to Section 6.3) (x (t)) 2 +(y (t)) 2 dt.3 Polr Coordintes.4 Polr Coordintes. Grphing. Circles (r =constnt. r = cosθ r = sinθ) nd crdioids (r = (±cosθ) or r = (±sinθ)) nd the flowers like r 2 = sin3θ..5 Are in polr coordintes R 3 2. Distnce between points. β α 2 f(θ)2 dθ 2. Eqution of sphere centered t (4,-,3) nd rdius 8. 2.2 Vectors. Length nd direction. Unit vectors. Force, displcement nd velocity. 2.3 Dot product. Algebric nd geometric difinitions. Angle between vectors. 2.3 Component of b long : comp b = ( b)/. Projection of b long : proj b = b 2. 2.4 Cross product. Algebric nd geometric definitions. b is perpendiculr to nd b nd by the right hnd rule. Length is the re of the prllelogrm. 2.4 Triple vector product ( b c). Tke bsolute vlue of this rel number to get the volume of the prllelepiped determined by the 3 vectors. 2.4 b is rel number; b is vector. 2.5 Eqution of plne through three points P Q nd R. The norml is PQ PR. 2.5 PrmetricequtionofthelinethroughP(,b,c)ndQis x,y,z =,b,c +tpq. Here PQ = d,d 2,d 3 is the direction vector of the line nd we hve x = +td, y = b+td 2 nd z = c+td 3 if we write out the components. 2.5 The symmetric equtions of the sme line re x d = y b d 2 = z c d 3