Estimating Areas Consider the challenge of estimating the volume of a solid {(x, y, z) 0 z f(x, y), (x, y) }, where is a region in the xy-plane. This may be thought of as the solid under the graph of z = f(x, y) and above the plane region. We could try partitioning into a large number n of small, compact subregions i, i = 1, 2,..., n. For each region i, we could pick some point P i = (x i, y i ) i and estimate the volume of the portion of the region aboue i by f(x i, y i ) A i, where A i is the area of that region. We could then estimate the entire volume by n i=1 f(x i, y i ) A i. This is reminiscent of a iemann Sum and, amazingly enough, will be called a iemann Sum. Double Integrals If these eimann Sums in a reasonable sense approach a limit when the regions get very small in all directions, we call that limit the double integral of the function over the region and denote it by f(x, y) da or f(x, y) dx dy. Properties of Multiple Integrals In general, properties of ordinary integrals which make some sense hold for double integrals and for multiple integrals in general. The integral of a sum or difference is the sum or difference of the integrals. If one function is bigger than another, then its integral is bigger. If a function is smaller than some constant, then its integral is smaller than that constant times the area of the region over which the integration is done. If a region is partitioned into subregions, then the integral over the entire region is equal to the sum of the integrals over the subregions. Applications of Double Integrals da is equal to the area of. If f(x, y) > 0, then f(x, y) da is equal to the volume of the region below the graph of z = f(x, y) and above the region in the plane. We will come up with additional applications in the near future. Calculation of a Double Integral as an Iterated Integral 1
The calculation of most double integrals reduces to repeated ordinary integration. Type I egion: A Type I egion is a plane region bounded on the left and right by vertical lines and above and below by the graphs of functions, which we will generally assume to be continuous. We will for the present describe it as {(x, y) α(x) y β(x), a x b}. a < b would be real numbers and α(x) < β(x) would be functions. Suppose we let P = {x 0, x 1, x 2, x 3,..., x n } be a partition of [a, b] and let i be the portion of for which x i 1 x x i. We can then calculate f(x, y) da = n i=1 i f(x, y) da. Partition and iemann Sum For a fixed i, we can estimate i f(x, y) da by choosing some x i 1 x i x i, forming a partition {y 0, y 1, y 2,..., y m } of the interval [α(x i ), β(x i )], choosing y j 1 yj y j for each j, and adding ( m m ) j=1 f(x i, yj ) x i y j = j=1 f(x i, yj ) y j x i. The expression in the parentheses is actually a iemann Sum which will be close to β(x i ) α(x i ) f(x i, y) dy. We can think of this as g(x i ), where g(x) = β(x) f(x, y) dy. α(x) So i f(x, y) da g(x i ) x i, and thus f(x, y) da n i=1 g(x i ) x i. The Iterated Integral This is itself a iemann Sum, which will be close to b g(x) dx. a We thus get f(x, y) da ( b ) β(x) f(x, y) dy dx, which we usually write without the parentheses as a α(x) b β(x) f(x, y) dy dx. a α(x) This is called an iterated integral. We calculate it by first treating x as a constant and calculating β(x) f(x, y) dy. This gives us a function α(x) of x, which we then integrate with respect to x, from a to b. Iterated Integrals Over Type II egions A region bounded above and below by horizontal lines and bounded on the left and right by curves which are the graphs of functions where y is the independent variable and x is the dependent variable is called a Type II egion. It may be described as {(x, y) α(y) x β(y), a y b}. 2
An integral over such a region may also be calculated as an iterated integral, for which we integrate with respect to x first and then y. If is such a region, then f(x, y) da = b β(y) f(x, y) dx dy. a α(y) Fubini s Theorem It s natural to ask, if a region is both a Type I egion and a Type II egion, whether we will obtain the same result regardless of which order we integrate. Fubini s Theorem says we do. We will not state Fubini s Theorem in any careful manner, but for now will be satisfied with the above vague statement. Double Integrals Using Polar Coordinates Suppose we wish to integrate some function over a region best described using polar coordinates. Such a region might be {(r, θ) α(θ) r β(θ), a θ b}. The region might be divided into subregions between two lines going through the origin, making an angle θ with each other, and two circles of radius approximately r for which the difference between the radii is approximately θ. These regions would be almost rectangular, and their areas might be estimated by multiplying together the length of one of their arcs, representing the base of the rectangle, and r, representing the height of the rectangle. The arcs are arcs in circles of radius r, subtendend through an angle θ, and thus has length r θ. We can thus approximate the area A of the rectangle by ( r) (r θ) = r r θ. This suggests replacing da in the integral, which is replaced by dx dy when using rectangular coordinates, with r dr dθ. We conclude, although we haven t actually proven it, that using polar coordinates and iterated integrals, α(θ) r β(θ), f(r, θ) da = b β(θ) f(r, θ)r dr dθ. a θ b a α(θ) This is sometimes written b dθ β(θ) dr rf(r, θ). a α(θ) Applications of Double Integrals Volume The volume of a solid of the form f(x, y) z g(x, y) for (x, y) is given by the double integral 3
g(x, y) f(x, y) da. Applications of Double Integrals Mass Suppose we, somewhat artificially, associate mass with a plane region by assigning a density δ = δ(x, y) with each point in the region. If we have a small, compact portion of the region with area A, we might approximate its mass by δ(x, y ) A. using some point (x, y ) in that portion of the region. We may thus estimate the entire mass by a iemann Sum δ(x i, y i ) A i, so we conclude the mass is δ(x, y) da. Applications of Double Integrals Moments Suppose we have a plane region with density function δ = δ(x, y) and wish to find its moments about the two axes and its center of mass. First, let s estimate its moment M y about the y axis. This may later be used to estimate the x coordinate of its center of mass by taking M y M. If we have a small, compact portion of the region with area A, we might approximate its contribution to the total moment by multiplying its mass m by x, the signed distance of a point in that portion from the y axis. Since m δ(x, y ) A, its contribution to the total moment is x δ(x, y ) A. We may thus estimate M y by a iemann Sum x i δ(x i, yi ) A i, so we conclude the moment is M y = xδ(x, y) da. Similarly, M x = If the center of mass is (x, y), then x = M y M and y = M x M. yδ(x, y) da. Surface Area 4
Consider a surface z = f(x, y), (x, y) D for some region D 2 for which we d like to find the surface area. If we take a small piece D of the region D with area A and look at the portion of the surface above it, that portion of the surface, if it s small enough, will be relatively flat. It s surface area S could be estimated by the area of the portion of the tangent plane directly above D. The tangent plane makes some angle γ with the xy plane. We may view portion of the tangent plane as mirroring D but scaled in one direction by the factor sec γ. It thus seems reasonable that S (sec γ) A. γ will also equal the angle a normal n makes with a vertical line, so n k = n k cos γ = n cos γ and sec γ = n n k. We know < f x, f y, 1 > is a normal to the plane, so we may take n =< f x, f y, 1 >. Then n = fx 2 + fy 2 + 1 and n k = 1, so sec γ = fx 2 + fy 2 + 1, and S fx 2 + fy 2 + 1 A. If we divide the region D into compact subregions D i with areas A i, the surface area of the corresponding portion of the surface will be S i f 2 x + f 2 y + 1 A i. We may thus estimate the entire surface area S = n i=1 S i n i=1 f 2 x + fy 2 + 1 A i. This is a iemann Sum and we conclude the surface area is S = fx 2 + fy 2 + 1 da. D 5