Chapter 6: Applications of Double Integration
Section 6.2: Volume
If fx,y is nonnegative over a plane region R, then ∫∫Rfx,y dA gives the volume that is inside the right cylinder whose cross section is R, and that is beneath the surface defined by z=fx,y and the plane z=0.
If fx,y is not nonnegative over a plane region R, then ∫∫Rfx,y dA gives the volume above the plane z=0 but below the positive part of z=fx,y, diminished by the volume between z=0 and the negative part of the surface z=fx,y.
The volume that is inside the right cylinder whose cross section is the plane region R, and that is bounded above by the graph of z=fx,y and below by the graph of z=gx,y is given by the double integral ∫∫Rfx,y−gx,y dA.
The Maple tools available for computing a volume "over R" are those detailed in Section 6.1. In each case, the integrand becomes fx,y instead of the "1" that is used to compute the area of R.
In Examples 6.2.(1-8), calculate the volume of the region bounded above by the surface z=Fx,y and below by the plane z=0, if the surface is defined over the given region R. Except for Example 6.2.5, the region R appears in the corresponding example from Section 6.1.
F=x y; R is the finite region bounded by the graph of y=x 1−x and the x-axis.
See Example 6.1.1.
F=2 x2+3 y2; R is the finite region bounded by the graphs of x=y2 and y=3−2 x.
See Example 6.1.2.
F=2 x+3 y+1; R is the region bounded by the graphs of fx=sinx and gx=sin2 x on 0≤x≤π. See Example 6.1.3.
F=3 x2+2 y2+1; R is the region bounded by the graphs of fx=arctanx+1−1/2 and gx=sinx on the interval x∈0,π/2. See Example 6.1.4.
F=5−3 x2−2 y2; R is the interior of the triangle whose vertices are 0,0,1,0,0,1.
F=x−32 y−62/30; R is the interior of the triangle whose vertices are 1,3,7,4,5,9. See Example 6.1.6.
F=2−y−1/22; R is the region bounded by the graphs of 1, cosx, and y=x on 0≤x≤1. See Example 6.1.7.
F=2−x2−4 y2; R is the interior of the ellipse x2+4 y2=1. See Example 6.1.8.
Calculate the volume bounded by the ellipsoid 3 x2+5 y2+7 z2=1.
Find the volume that is enclosed by the surfaces z1=7−x2−y2 and z2=2 x+4 y−6 and that lies inside the right cylinder whose footprint in the plane z=0 is bounded by the curves y=x, y=2−x2, and x=0.
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