2CONTENTSOrdinary Differential Equations of First Order and First DegreeLinear Differential Equations of Second and Higher OrderMean Value TheoremsFunctions of Several VariablesCurvature, Evolutes and EnvelopesCurve TracingApplications of IntegrationMultiple IntegralsSeries and SequencesVector Differentiation and Vector OperatorsVector IntegrationVector Integral TheoremsLaplace transforms
3TEXT BOOKSA text book of Engineering Mathematics, Vol-I T.K.V.Iyengar, B.Krishna Gandhi and Others, S.Chand & CompanyA text book of Engineering Mathematics, C.Sankaraiah, V.G.S.Book LinksA text book of Engineering Mathematics, Shahnaz A Bathul, Right PublishersA text book of Engineering Mathematics, P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar Rao, Deepthi Publications
4REFERENCESA text book of Engineering Mathematics, B.V.Raman, Tata Mc Graw HillAdvanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd.A text Book of Engineering Mathematics, Thamson Book collection
5CHAPTER-I:APPLICATIONS OF INTEGRATION CHAPTER-II:MULTIPLE INTEGRALS UNIT-VCHAPTER-I:APPLICATIONS OF INTEGRATIONCHAPTER-II:MULTIPLE INTEGRALS
6UNIT HEADER Name of the Course: B.Tech Code No:07A1BS02 Year/Branch: I Year CSE,IT,ECE,EEE,ME,CIVIL,AEROUnit No: VNo. of slides:17
7UNIT INDEX UNIT-VS. No.ModuleLectureNo.PPT Slide No.1Introduction, Length, Volume and Surface areaL1-58-112Multiple integrals, Change of order of integrationL6-1012-153Triple integration , Change in triple integrationL11-1216-19
8Lecture-1 APPLICATIONS OF INTEGRATION Here we study some important applications of integration like Length of arc, Volume, Surface area etc.,RECTIFICATION: The process of finding the length of an arc of the curve is called rectification.Length of an arc S=∫[1+(dy/dx)2]1/2
9Lecture-2 LENGTH OF CURVE(RECTIFICATION) The process of finding the length of an arc of the curve is called rectification. We can find length of the curve in Cartesian form, Polar form and Parametric form.Length of curve in cartesian form: S=∫[1+(dy/dx)2]1/2Length of curve in parametric form:S=∫√(dx/dθ)2+(dy/dθ)2 dθ
10Lecture-3 ARC LENGTH Polar form: If r=f(θ) and θ=a, θ=b then S=∫√r2+(dr/dθ)2 dθIf θ=f(r) and r=r1 , r=r2 then S=∫√1+r2(dθ/dr)2 dr
11Lecture-4 VOLUMEIf a plane area R is revolved about a fixed line L in its plane, a solid is generated. Such a solid is known as solid of revolution and its volume is called volume of revolution. The line L about which the region R is revolved is called the axis of revolution.Volume of the solid can be found in 3 different forms Cartesian form, Polar form and Parametric form.Volume of the solid about x-axis=∫пy2dx
12Lecture-5 FORMULAE FOR VOLUME Cartesian form: Volume of the solid about x-axis=∫пy2dx Volume of the solid about y-axis=∫пx2dyVolume of the solid about any axis=∫п(AR)2d(OR)Volume bounded by two curves= ∫п(y12-y22)dx
13Lecture-6 SURFACE AREAThe surface area of the solid generated by the revolution about the x-axis of the area bounded by the curve y=f(x).We can find revolution about x-axis,y-axis,initial line, pole and about any axis.Example: The Surface area generated by the circle x2+y2=16 about its diameter is 64π
14Lecture-7 MULTIPLE INTEGRALS Let y=f(x) be a function of one variable defined and bounded on [a,b]. Let [a,b] be divided into n subintervals by points x0,…,xn such that a=x0,……….xn=b. The generalization of this definition ;to two dimensions is called a double integral and to three dimensions is called a triple integral.
15Lecture-8 DOUBLE INTEGRALS Double integrals over a region R may be evaluated by two successive integrations. Suppose the region R cannot be represented by those inequalities, and the region R can be subdivided into finitely many portions which have that property, we may integrate f(x,y) over each portion separately and add the results. This will give the value of the double integral.
16Lecture-9 CHANGE OF VARIABLES IN DOUBLE INTEGRAL Sometimes the evaluation of a double or triple integral with its present form may not be simple to evaluate. By choice of an appropriate coordinate system, a given integral can be transformed into a simpler integral involving the new variables. In this case we assume that x=r cosθ, y=r sinθ and dxdy=rdrdθ
17Lecture-10 CHANGE OF ORDER OF INTEGRATION Here change of order of integration implies that the change of limits of integration. If the region of integration consists of a vertical strip and slide along x-axis then in the changed order a horizontal strip and slide along y-axis then in the changed order a horizontal strip and slide along y-axis are to be considered and vice-versa. Sometimes we may have to split the region of integration and express the given integral as sum of the integrals over these sub-regions. Sometimes as commented above, the evaluation gets simplified due to the change of order of integration. Always it is better to draw a rough sketch of region of integration.
18Lecture-11 TRIPLE INTEGRALS The triple integral is evaluated as the repeated integral where the limits of z are z1 , z2 which are either constants or functions of x and y; the y limits y1 , y2 are either constants or functions of x; the x limits x1, x2 are constants. First f(x,y,z) is integrated w.r.t. z between z limits keeping x and y are fixed. The resulting expression is integrated w.r.t. y between y limits keeping x constant. The result is finally integrated w.r.t. x from x1 to x2.
19Lecture-12 CHANGE OF VARIABLES IN TRIPLE INTEGRAL In problems having symmetry with respect to a point O, it would be convenient to use spherical coordinates with this point chosen as origin. Here we assume that x=r sinθ cosф, y=r sinθ sinф, z=r cosθ and dxdydz=r2 sinθ drdθdфExample: By the method of change of variables in triple integral the volume of the portion of the sphere x2+y2+z2=a2 lying inside the cylinder x2+y2=ax is 2a3/9(3π-4)