# Multiple Integral

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chapter multiple integrals double integrals, iterated

. Taylor’s Theorem. Suppose f (x, y) and its partial derivatives through order n + 1 are continuous throughout an open rectangular region R centered at a point (a, b). Then, throughout R, 1 = f (a, b) + (hfx + kfy )|(a,b) + 2! (h2 fxx + 2hkfxy + k2 fyy )|(a,b) 1 3f 2 kf + 3! (h xxx + 3h xxy + 2hk2 fxyy + k3 fyyy )|(a,b) + · · · ( )n ( )n+ ∂ ∂ ∂ ∂ + n! h ∂x + k ∂y f |(a,b) + (n+1)! h ∂x + k ∂y f |(a+ch,b+ck) for some c ∈ (0, 1)RemarksThe proof just applies the chain rule and the trick of n-th Taylor 1 .chapter multiple integration

Consider a surface f (x, y); you might temporarily think of this as representing physical topography—a hilly landscape, perhaps. What is the average height of the surface (or average altitude of the landscape) over some region? As with most such problems, we start by thinking about how we might approximate the answer. Suppose the region is a rectangle, [a, b] × [c, d]. We can divide the rectangle into a grid, m subdivisions in one direction and n in the other, as indicated in ﬁgure 15.1. We pick x values .chapter multiple integrals double integrals, iterated

. Example Find the surface area of the torus generated by revolving the circle C : (x − a)2 + z2 = a2 (0 < a < b) in the xy-plane around the z-axisSolution. For any point P(x, 0, z) lying on the circle C, parameterize P by z = a sin ψ and x − a = b cos ψ, where 0 ≤ ψ ≤ 2π. As one rotates the vector OP about the z-axis with angle from 0 to 2π, then the locus of the moving point P will trace out a circle of radius ℓ = ∥OP∥ = b + a cosIn this case, we ﬁrst note that z-coordinate of the moving point from.
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