Chapter 20

August 21 2017

Chapter 20 Multiple Integrals 20.1 Volumes as Iterated Integrals - a continuous function f(x, y) of two variables can be integrated over a plane region R - in much the same way a continuous function of one variable can be integrated over an interval - the result is a number called the double integral of f(x, y) over R - SS R f(x, y) dA or SS R f(x, y) dx dy - a different but clearly related concept is that of an iterated integral - we discuss iterated integrals in this section - in the next section we return to the topic of double integrals - and explain what they are and how they are related to iterated integrals - in section 7.3 we discussed the "method of moving slices" - for finding volumes - V = Sab A(x) dx - dV = A(x) dx - is the volume of a thin slice of the solid of thickness dx - the total volume is found by - adding together (or integrating) - these elements of volume as our typical slice sweeps through the complete solid - however - if the section itself has curved boundaries - then the determination of A(x) also requires integration - where y1(x) and y2(x) are the equations of the curves that bound the base on the left and right - to find the total volume - we insert the dA equation into the dV one - to obtain the iterated integral - V = Sab [ S y1(x) y2(x) f(x, y) dy ] dx - notice we first integrate f(x, y) with respect to y - holding x fixed - the limits of integration depend on the fixed but arbitrary value of x - so does the resulting value of the inner integral - the inner integral is precisely the function of A(x) - which we then integrate with respect to x from a to b to obtain the volume - to summarize - we start with a positive function f(x, y) of two variables - we first "integrate y out" which gives a function of x alone - then we "integrate x out" which gives a number, the volume of the solid - in some cases - it may be more convenient to cut the solid by a plane perpendicular to the y-axis - and form the integrated integral in the other order - first integrating x and then y 20.2 Double Integrals and Iterated Integrals - the double integral of a function of two variables is the two-dimensional analog of the definite integral of a function of one variable - the value of the single integral Sab f(x) is determined by f(x) and the interval [a, b] - in the case of the double integral - the interval [a, b] is replaced by a region R in the xy-plane - SS R f(x, y) dA, dA instead of dx,dy - in section 6.4 a single integral was defined as the limit of certain sums - we define the double integral the same way - SS R f(x, y) dA = lim E k1n f(xk, yk) dAk - each f(xk, yk) dAk is approximately the volume of a thin column - when the region R has a certain simple shape the double integral is always equal to a suitably chosen iterated integral - if R is vertically simple then - SS R f(x, y) dA = Sab S y1(x) y2(x) dy dx - a double integral is a number associated with a function f(x, y) and a region R - this number exists and has a meaning independently of any particular method of computing it - on the other hand - an iterated integral is a double integral plus a built-in computational procedure - so every iterated integral is a double integral - but not vice versa 20.3 Physical Applications of Double Integrals - we have seen - SS R f(x, y) dA - gives the volume of a certain solid - we think of the volume as composed of infinitely many thin columns - each standing on an infinitely small rectangular element of area dA - whose sides are dx and dy - dA = dx dy = dy dx - so its volume is - dV = f(x, y) dA - the two ways of calculating the volume as an iterated integral - are first to allow dA to move across R along a thin horizontal strip - integrating first x then y - or to move dA across R along a thin vertical strip - depending on which iterated integral we wish to consider - this description of the intuitive meaning of the double integral - expresses the essence of the Leibniz approach to integration - to find the whole of the quantity - imagine it to be judiciously divided into a great many small pieces - then add these pieces together - this is the unifying theme of the applications of double integrals - the integral has many other useful interpretations that arise by making special choices of the function f(x, y) - for example - the mass, moment, center of mass, moment of inertia, polar moment of inertia - for all of these - we obtain the total quantity under discussion by adding together - the "infinite small" parts of it associated with the element of area dA - as dA sweeps over the region R 20.4 Double Integrals in Polar Coordinates - it is often more convenient to describe the boundaries of a region by using polar coordinates - in these circumstances we can usually save a lot of work by expressing the double integral in terms of polar coordinates - the double integral can be given an equivalent definition by meals of small "polar rectangles" - dA = (dr) (r d angle) = r dr d angle - SS R f(x, y) dA = SS R f(r cos angle, r sin angle) r dr d angle - if the region R is radially simple - we can use the iterated integral - SS R f(x, y) dA = S ab S r1(angle) r2(angle) f( r cos angle, r sin angle) r dr d angle 20.5 Triple Integrals - the definition of a triple integral follows the same pattern of ideas that were used to define a double integral - a triple integral involves a function f(x, y, z) defined on a three-dimensional region R - we divide R into many small rectangular boxes - SSS R f(x, y, z) dV = lim E f(xk, yk, zk) dVk = SSS R f(x, y, z) dx dy dz 20.6 Cylindrical Coordinates - if a solid has axial symmetry - it is often convenient to place its axis of symmetry on the z-axis - and use cylindrical coordinates - dV = r dr d angle dz 20.7 Spherical Coordinates. Gravitational Attraction - spherical coordinates are designed to fit situations with symmetry about a point 20.8 Areas of Curved Surfaces - in section 7.6 we discussed the problem of finding the area of a surface of revolution - we now consider the area problem for more general curved surfaces - the method rests on the simple fact that if two planes intersect at an angle y - then all arsa in one plane are multiplied by cos y when projected on the other - A = S cos y - so S = SS dS = SS R dA / cos y