ENGINEERING MATH II
ENGINEERING MATH II MATH 152
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Date Created: 10/21/15
MATH 152 Fall 2008 Yuliya Gorb gorb math tamu edu wwwmathtamuedu gorbmath1521a112008html Formula Sheet for Exam 2 Approximate Integration In all cases partition the given interval a 12 into n subintervals With endpoints a 10 lt 11 lt lt In I Let the size of equal subintervals of the partition be bia 77 n AI hence7 1k a kAz o Midpoint Rule Let 1 11 E 11671 1k be the midpoint of the subinterval 116717116 1 Then the rule is b fltzgtdz Ax W1 ms t i i m 2 The error of the approximation is Kb 7 a3 WM 3 W 7 Where S K for a S I S b o Trapezoidal Rule 1 The rule is b mm Waco 2fI1 2fltz2gt 2fltzmgt my 2 The error of the approximation is lETl S 7 Kbia3 12n2 Where S K for a S I S b o Simpson s Rule 1 The rule is b mm Waco mm 2M2 W3 210m mm mm Where n is even 2 The error of the approximation is K b 7 a 5 lEsl 180n Where lf4zl S K for a S I S b Improper Integrals 1 type I 7 In nite Intervals B o If exists for every B 2 a then 00 B Blim fzdz provided this limit exists as a nite number hence the improper integral is called convergent otherwise 7 divergent b o If exists for every A S b then A b 1 00mm 1133100 A rm provided this limit exists as a nite number hence the improper integral is called convergent otherwise 7 divergent o If both and are convergent then x mm fzdzamfzdm 2 type II 7 Discontinuous Integrands o If f is continuous on ab and is discontinuous at z b then b B Af1dIBll gtI 17a fzdz provided this limit exists hence the improper integral is called convergent otherwise 7 diver gent o If f is continuous on ab and is discontinuous at z a then b b AfzdzAlirnA fzdz provided this limit exists hence the improper integral is called convergent otherwise 7 diver gent C b o If f is discontinuous at z c where a lt c lt b and both and are convergent then b c b a a 0 Differential Equations of the rst order 1 Separable Di erential Equations written in the form 7 fltzgtgltygt can be solved by the separation of variables gltygt f 0 forgltygt ol 2 Linear Di erential Equations To solve the normalized linear differential equation of the form y 01y 4I7 1 rst nd an integrating factor 101 61mm and then multiply both sides of 1 by and integrate 3 Application of Di erential Equations to eg mixing problem A tank of the capacity of V liters initially contains V0 L of brine With Q0 g of salt in it The brine containing cl gL of salt enters the tank at the rate of 7 1 Lmin and wellstirred mixture leaves the tank at the rate of 7 2 lt 7 1 Lmin Find the amount of salt in the tank at the moment When it starts over owing Solution a Find the volume of the brine at any moment of time V05 V0 T1 7 T2t L b Let be an amount of salt at time t Then Q0 g The rate of change derivative of of the amount of salt in the tank is equal to the difference of the ratein Tin and rateout Tout of salt dQ E Tin Tout c Ratein of salt is the product of the rate at Which the brine enters the tank and the concentration of salt in that amount Tin T161 d Rateout of salt is the product of the rate at Which the brine leaves the tank and the concentration t of salt in that amount Tout T102 Where Cg gL e Hence7 the differential equation With the initial condition that govern this process are dQ QW E T2 V 7 T1017 Q0 Q0 g which is to be solved With respect to f Let T be moment of time When tank starts over owing Then Vquot 7 V VT V that is V0 T1 7 TM V or T 7 7 1 7 7 2 g Then the amount of salt at the moment When the tank starts over owing is equal to the function Q found in e evaluated at T found in f and measured in g Arc Length 1 The length if a smooth curve given parametrically by Ift7 ygt agtgby and traversed only once as t runs from a to b is 2 1f the curve is given by y for a S I S b then its length is b 2 L 1lt gt dzi a dz 3 1f the curve is given by z gy7 for c S y S d then its length is d 2 L 1 dyi c V dy 1 The area of the surface generated by rotating the curve y a S I S b about the zaxis is Area of a Surface of Revolution 1 A 27rfz 1fIl2dzv b 2 A 27ry 1 dzi a dz 1 The area of the surface generated by rotating the curve I gy7 c S y S d about the zaxis is 1 d1 2 A 27ry 1 dzi c dy or equivalently7 to 3 The area of the surface generated by rotating the curve y a S I S b about the yaxis is b 2 d A 27m 1ltygt dzi a dz 4 The area of the surface generated by rotating the curve I gy7 c S y S d about the yaxis is d 2 d A 27m 1ltigt dzi c V dy 5 ln general7 the area of the surface of the revolution generated by revolving a curve 0 about the zaxis is S 27ryds 0 about the yaxis is S 27rzds Where d3 I2 t y 2tdti Moments and Center of Mass Centroid 1 Let the plate of density p is given by the region R is bounded by the curve y a S I S 12 o The mass of the plate R is m pab o The moment of R about the ziaxis is I My p gimrdm o The moment of R about the yiaxis is I My p o The centroid center of mass of R is located at the point 1 y where 1 My ffzfltzgtdz m f 7 jammy m f y 2 Let the plate R is bounded by two curves y y 91 a S I S 127 where 2 Then the centroid of R is located at the point 1 y where 1 aw i Ifryzldz b a i am 7 gm dz where A is the area between the two curves7 ie I A W 79mm Hydrostatic Pressure and Force A plate with area A is submerged in a uid of density p at a depth d below the surface of the uid Then the force exerted by the uid on the plate is F mg pVg pgAdi The pressure on the plate is F P 7 d A P9 In British units the pressure is P pgd 6d where 6 is the weight density of the uid in lbft3i