About the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a three-year curriculum development project undertaken by a consortium of five English universities led by Loughborough University, funded by the Higher Education Funding Council for England under the Fund for the Development of Teaching and Learning for the period October 2002 – September 2005, with additional transferability funding October 2005 – September 2006. HELM aims to enhance the mathematical education of engineering undergraduates through flexible learning resources, mainly these Workbooks. HELM learning resources were produced primarily by teams of writers at six universities: Hull, Loughborough, Manchester, Newcastle, Reading, Sunderland. HELM gratefully acknowledges the valuable support of colleagues at the following universities and colleges involved in the critical reading, trialling, enhancement and revision of the learning materials: Aston, Bournemouth & Poole College, Cambridge, City, Glamorgan, Glasgow, Glasgow Caledonian, Glenrothes Institute of Applied Technology, Harper Adams, Hertfordshire, Leicester, Liverpool, London Metropolitan, Moray College, Northumbria, Nottingham, Nottingham Trent, Oxford Brookes, Plymouth, Portsmouth, Queens Belfast, Robert Gordon, Royal Forest of Dean College, Salford, Sligo Institute of Technology, Southampton, Southampton Institute, Surrey, Teesside, Ulster, University of Wales Institute Cardiff, West Kingsway College (London), West Notts College. HELM Contacts: Post: HELM, Mathematics Education Centre, Loughborough University, Loughborough, LE11 3TU. Email: [email protected] Web: http://helm.lboro.ac.uk HELM Workbooks List 1 Basic Algebra 26 Functions of a Complex Variable 2 Basic Functions 27 Multiple Integration 3 Equations, Inequalities & Partial Fractions 28 Differential Vector Calculus 4 Trigonometry 29 Integral Vector Calculus 5 Functions and Modelling 30 Introduction to Numerical Methods 6 Exponential and Logarithmic Functions 31 Numerical Methods of Approximation 7 Matrices 32 Numerical Initial Value Problems 8 Matrix Solution of Equations 33 Numerical Boundary Value Problems 9 Vectors 34 Modelling Motion 10 Complex Numbers 35 Sets and Probability 11 Differentiation 36 Descriptive Statistics 12 Applications of Differentiation 37 Discrete Probability Distributions 13 Integration 38 Continuous Probability Distributions 14 Applications of Integration 1 39 The Normal Distribution 15 Applications of Integration 2 40 Sampling Distributions and Estimation 16 Sequences and Series 41 Hypothesis Testing 17 Conics and Polar Coordinates 42 Goodness of Fit and Contingency Tables 18 Functions of Several Variables 43 Regression and Correlation 19 Differential Equations 44 Analysis of Variance 20 Laplace Transforms 45 Non-parametric Statistics 21 z-Transforms 46 Reliability and Quality Control 22 Eigenvalues and Eigenvectors 47 Mathematics and Physics Miscellany 23 Fourier Series 48 Engineering Case Study 24 Fourier Transforms 49 Student’s Guide 25 Partial Differential Equations 50 Tutor’s Guide © Copyright Loughborough University, 2015 Production of this 2015 edition, containing corrections and minor revisions of the 2008 edition, was funded by the sigma Network. ContentsContents 2727 Multiple Integration 27.1 Introduction to Surface Integrals 2 27.2 Multiple Integrals over Non-rectangular Regions 20 27.3 Volume Integrals 41 27.4 Changing Coordinates 66 Learning outcomes In this Workbook you will learn to integrate a function of two variables over various rectangular and non-rectangular areas. You will learn how to do this for various other coordinate systems. You will learn to integrate a function of three variables over a volume. Introduction to Surface Integrals 27.1 Introduction Often in Engineering it is necessary to find the sum of a quantity over an area or surface. This can be achieved by means of a surface integral also known as a double integral i.e. a function is integrated twice, once with respect to one variable and subsequently with respect to another variable. This Section looks at the concept of the double integral and how to evaluate a double integral over a rectangular area. thoroughly understand the various techniques # • Prerequisites of integration be familiar with the concept of a function of Before starting this Section you should ::: • two variables " understand the concept of a surface integral ! Learning Outcomes • integrate a function over a rectangular region On completion you should be able to ::: • 2 HELM (2015): Workbook 27: Multiple Integration ® 1. An example of a surface integral An engineer involved with the construction of a dam to hold back the water in a reservoir needs to be able to calculate the total force the water exerts on the dam so that the dam is built with sufficient strength. In order to calculate this force, two results are required: (a) The pressure p of the water is proportional to the depth. That is p = kd (1) where k is a constant. (b) The force on an area subjected to constant pressure is given by force = pressure area (2) × The diagram shows the face of the dam. The depth of water is h and δA is a small area in the face of the dam with coordinates (x; y). y h y δA − h y x x Figure 1 Using (1), the pressure at δA k(h y). Using (2), the force on an area δA k(h y)δA. Both of these expressions are approximate∼ − as y is slightly different at the top of∼δA to− the bottom. Now Total force on dam = sum of forces on all areas δA making up the face of the dam X k(h y)δA ≈ − all δA For a better approximation let δA become smaller, and for the exact result find the limit as δA 0. Then ! X Total force on the dam = lim k(h y)δA δA!0 − Z = k(h y) dA A − Z where k(h y) dA stands for the surface integral of k(h y) over the area A. Surface integrals A − − are evaluated using double integrals. The following Section shows a double integral being developed in the case of the volume under a surface. HELM (2015): 3 Section 27.1: Introduction to Surface Integrals 2. Single and double integrals As has been seen in 14.3, the area under the curve y = f (x) between x = a and x = b is Z b given by f(x) dx (assuming that the curve lies above the axis for all x in the range a x b). a ≤ ≤ This is illustrated by the figure below. y area required y = f(x) a b x Figure 2 In a similar manner, the volume under a surface (given by a function of two variables z = f(x; y)) and above the xy plane can be found by integrating the function z = f(x; y) twice, once with respect to x and once with respect to y: f(x, y) y y = d y = c x x = a x = b Figure 3 The above figure shows the part of a surface given by f (x; y) which lies above the rectangle a x b, c y d. This rectangle is shaded and the volume above this rectangle but below the≤ surface≤ can≤ be seen.≤ 4 HELM (2015): Workbook 27: Multiple Integration ® f(x, y) δx y y = d y = c x x = a x = b Figure 4 Imagine a vertical slice taken through this volume at right angles to the x-axis (figure above). This slice has thickness δx and lies at position x. Assuming that δx is small enough that the areas of both sides (left and right) of this slice are virtually the same, the area of each face of the slice is given by the integral Z y=d f (x; y) dy (where x measures the position of the slice) y=c and the volume of the slice will be given by Z y=d δx f (x; y) dy y=c To find the total volume between the surface and the xy plane, this quantity should be summed over all possible such slices, each for a different value of x. Thus X Z y=d V f (xi; y) dy δx ≈ i y=c When δx becomes infinitesmally small, it can be considered to be dx and the summation will change into an integral. Hence Z x=b Z y=d V = f (x; y) dydx x=a y=c Thus the volume is given by integrating the function twice, once with respect to x and once with respect to y: The procedure shown here considers the volume above a rectangular area and below the surface. The volume beneath the surface over a non-rectangular area can also be found by integrating twice (see Section 27.2). HELM (2015): 5 Section 27.1: Introduction to Surface Integrals Key Point 1 Volume Integral The volume under the surface z = f(x; y) and above a rectangular region in the xy plane (that is the rectangle a x b; c y d) is given by the integral: ≤ ≤ ≤ ≤ Z b Z d V = f(x; y) dydx x=a y=c 3. ‘Inner’ and ‘Outer’ integrals A typical double integral may be expressed as Z x=b Z y=d I = f (x; y) dy dx x=a y=c where the part in the centre i.e. Z y=d f (x; y) dy y=c (known as the inner integral) is the integral of a function of x and y with respect to y: As the integration takes place with respect to y, the variable x may be regarded as a fixed quantity (a constant) but for every different value of x, the inner integral will take a different value. Thus, the Z y=d inner integral will be a function of x e.g. g (x) = f (x; y) dy. y=c This inner integral, being a function of x, once evaluated, can take its place within the outer integral Z x=b i.e. I = g (x) dx which can then be integrated with respect to x to give the value of the x=a double integral.