ENGINEERING MATH II
ENGINEERING MATH II MATH 152
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This 7 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 152 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/226040/math-152-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Date Created: 10/21/15
MATH 152 Spring 2008 Yuliya Gorb goermath tamu edu wwwmathtamu edu gorbmath1522008 html Formula Sheet for Exam 2 Approximate Integration In all cases partition the given interval ab into n subintervals with end points a 10 lt 11 lt lt In I Let the size of equal subintervals of the partition be I 7 a AI 7 n hence7 1k a kAz o Midpoint Rule Let 1 11c 1k71 1k be the midpoint of the subinterval 11617 1 Then the rule is b fltzgtdz Ax fzi ms i i i M 2 The error of the approximation is K02 7 a3 lt 7 EM l 24n2 Where S K for a S I S b o Trapezoidal Rule 1 The rule is b mm Waco 2M 2M2 mm may 2 The error of the approximation is K02 7 a3 lt 7 ET S 12 2 Where S K for a S I S b o Simpson s Rule H i The rule is b mm Waco mm 2M2 W3 i i i 210m mm rm where n is even to i The error of the approximation is Kb7 a5 E lt lsli 180n4 where f4zl S K for a S I S bi 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 If exists for every A S b then A b b 0016de 1133100 A rm provided this limit exists as a nite number hence the improper integral is called convergent otherwise 7 divergent If both and are convergent then x mm fzdzofzdzi 2 type II 7 Discontinuous Integrands o If f is continuous on ab and is discontinuous at z b then b B a Maggi mm provided this limit exists hence the improper integral is called con vergent otherwise 7 divergent i o If f is continuous on a7 12 and is discontinuous at z a7 then b b AfzdzAlimAfzdz provided this limit exists hence7 the improper integral is called con vergent7 otherwise 7 divergent o If f is discontinuous at z c Where a lt c lt 127 and both b and are convergent7 then c Ab x x mum mz Differential Equations of the rst order 1 Separable Di erential Equations Written in the form jg fltzgtgltygt can be solved by the separation of variables dy 7 1 dz C7 99 fl for M y 0 Linear Di erential Equations E0 To solve the normalized linear differential equation of the form y pry 41 1 rst nd an integrating factor 101 61mm and then multiply both sides of l by and integrate 9 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 Wt V0 n 7 mt 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 7 Tin Tout dt 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 of salt in that amount Tout T102 Where Qt gL Wt e Hence7 the differential equation With the initial condition that govern this process are 52 T23Eg 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 V0 T17T2 VTV7 thatisy V0T1T2TV7 or T 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 y9t7 aStSb and traversed only once as t runs from a to b is ltgt2ltigt 2 If the curve is given by y for a S I S b then its length is b 2 L llt gt dz a dz 3 1f the curve is given by z gy7 for c S y S d then its length is d 2 L 1ltdigt dyi c V dy Area of a Surface of Revolution 1 The area of the surface generated by rotating the curve y a S I S b about the zaxis is b A 27rfz 1fIl2dzv b 2 A 27ry 1 dzi a V dz or equivalently7 2 The area of the surface generated by rotating the curve I gy7 c S y S d about the zaxis is d 2 d A 27ry 1 dzi c V dy 3 The area of the surface generated by rotating the curve y a S I S b about the yaxis is b 2 A 27m 1 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 A 27m 1 dzi c V dy Moments and Center of Mass Centroid 1 Let the plate of density p is given by the region R is bounded by the curve yfzaSzltb o The mass of the plate R is m pzb o The moment of R about the ziaxis is b 1 MT p Emach 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 1yquot where b 1 fa xfltzgtdz m f 7 b fa mmw m f 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 W ere y 1 av Ifryzldz b y g u z tma m where A is the area between the two curves7 ie I A W 7 yltzgtldm 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 Review Problems for Sections 84 95 X31 X3 X2 84 Compute f dX 88 Use the Trapezoidal rule to approximate indx with n 5 Compute the x error ofthis approximation you will not need to memorize the error bound formulas but you will be responsible for being able to apply them 89 State the comparison theorem for improperintegrals Using this theorem mlnX determine whether f3 73dx converges or diverges X 91 Solve the differential equation y ysin X Solve the initial value problem for this equation with y0 1 92 Solve the differential equation xy 3y X2 First compute the integrating factor then use the general formula 93 Find the arc length ofthe parameterized curve 1 t2 1t3 between the points 11 and 22 94 The arc of the parabola y X2 from 00 to 11 is rotated about the yaxis Find the area ofthe resulting surface 95 Find the centroid ie center of mass ofthe region bounded by the line y X and the parabola y X2
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