**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