Chapter 5: Double Integration
Section 5.1: The Double Integral
Table 5.1.1 makes a quick comparison between the definite integral in one variable, and the (definite) double integral in two variables. Each definition in the table is subject to conditions such as "over all partitions whose norm goes to zero". The limit in the case of the double integral must be the bivariate limit, not an iterated limit.
limm,n→∞,∞∑i=1n ∑j=1mfxij,yijΔ Aij
Signed area between the graph of fx and the x-axis
Signed volume between the graph of fx,y and the xy-plane
Table 5.1.1 Comparison of the single and double integral
For the double integral, the region R is initially taken as the rectangle a≤x≤b,c≤y≤d, which is partitioned into a grid of sub-rectangles. An evaluation point xij,yij is chosen in each sub-rectangle and the function evaluated at that point. The double Riemann sum is the sum of products of such function values times Δ Aij, the area of the related sub-rectangle.
Because of the Fundamental Theorem of Calculus, evaluation of the definite (single) integral amounts to finding and evaluating an antiderivative, the indefinite integral symbol ∫ is taken to mean "antiderivative of." There seems to be little call for an indefinite double integral; it will be mentioned only if absolutely necessary in an application.
The signed area between the graph of fx and the x-axis is the sum of area above the axis taken as positive, and area below the axis taken as negative. Hence, it is the "area beneath fx" only when fx is nonnegative. Computing actual area bounded by a curve that crosses the x-axis requires knowing the x-intercept. Sometimes Maple can figure this out and integrating fx succeeds. Otherwise, the integral has to be split into two parts.
The signed volume between the graph of fx,y and the xy-plane is the sum of volume above the plane z=0 taken as positive, and volume below the plane z=0 taken as negative. Hence, it is the true volume only when fx,y is nonnegative. Computing actual volume bounded by a surface that crosses the plane z=0 requires knowing the bounds of the regions over which f is positive, and negative, and integrating separately over these regions.
Apply the definition of the double integral to fx,y=x2+y2 on the square 0≤x,y≤1.
Apply the definition of the double integral to obtain the volume inside the cylinder bounded by the planes x=±1,y=±1/2, but between the surface fx,y=7−3 x2−5 y2 and the xy-plane.
Apply the definition of the double integral to obtain the volume bounded by the surface fx,y=1+5 x2+7 y2, the xy-plane, and the planes x=±1,y=±2.
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