Chapter 8: Applications of Triple Integration
Section 8.1: Volume
The examples in Section 7.3, Section 7.4, and Section 7.6 illustrated how to compute volumes with iterated triple integrals in Cartesian, cylindrical, and spherical coordinates, respectively.
The examples in this section require the calculation of volumes of regions R described in such a way as to require a bit more visualization skill, the translation into appropriate mathematical notation of the words used to detail shapes and other geometric objects.
In Examples 8.1.(1 - 30), use an iterated triple integral to obtain the volume of the given region R.
R is the tetrahedron bounded by the planes x+3 y+z=5, x=3 y, x=0, z=0.
R is the tetrahedron cut from the first octant by the plane 3 x+5 y+7 z=15.
R is the region interior to the cylinder x2+y2=9 that is bounded below by the xy-plane, and above by the paraboloid z=x2+y2.
R is the first-octant region bounded by the planes x+z=3, and y+5 z=15.
R is the wedge the planes z=y and z=0 cut from the cylinder x2+y2=4.
R is the first-octant region bounded above by the cylinder z=3−x2 and on the right by the paraboloid 3 y=x2+z2.
R is the region enclosed by the cylinder y2+4 z2=16 and the planes x=1 and x+y=5.
R is the region common to the two cylinders x2+y2=1 and x2+z2=1.
R is the region enclosed by the cylinders z=4−x2 and z=5 x2, and the planes y=0 and x+y=2/3.
R is the region that is inside the cylinder x2+4 y2=4, and that is bounded above and below by the planes z=x+2, and z=0, respectively.
R is the region bounded by the elliptic paraboloids z=2 x2+5 y2 and z=25− 4 x2−10 y2.
R is the region in the upper half-plane that is above the cone z2=x2+y2 but below the sphere x2+y2+z2=18. (Use cylindrical coordinates.)
R is the region in the upper half-plane that is above the cone z2=x2+y2 but below the sphere x2+y2+z2=18. (Use spherical coordinates.)
If ρ,φ,θ are the variables in spherical coordinates, R is the region bounded inside by the surface ρ=1+cosφ and outside by the sphere ρ=2.
R is the region inside the cylinder x2+y2=9 and between the planes z=0 and y+z=5.
R is the region that lies inside the sphere x2+y2+z2=4, and is between the cones z=x2+y2 and z=3x2+y2.
R is the region that is bounded above by the paraboloid z=4−x2−y2, below by the plane z=0, and that lies outside the cylinder x2+y2=1.
R is the region bounded below by the paraboloid z=x2+y2 and above by z=4 x.
R is the first-octant region that lies between the cylinders r=1 and r=3, and that is bounded below by the plane z=0 and above, by the surface z=1+ x y.
R is the region that lies between the plane z=0 and the paraboloid z=9−x2−y2.
R is the region enclosed by the surface ρ=1−cosφ, where ρ,φ,θ are the variables in spherical coordinates.
R is the region that lies between the paraboloids z=4−x2−y2 and z=3 x2+3 y2.
R is the region that lies inside both the sphere x2+y2+z2=16 and the cylinder x2+y2−4 y=0.
R is the region bounded above by the surface z=10 y, and below by the surface z=2 x2+3 y2.
R is the region bounded above by the sphere x2+y2+z2=6, and below by the paraboloid z=x2+y2.
R is the region interior to the surface ρ=2 sinφ, where ρ,φ,θ are the variables of spherical coordinates.
R is the first-octant region enclosed by the cylinder x2+z2=4 and the plane y=3.
R is the region bounded by the paraboloid z=4−x2−y2 and the plane z=0.
R is the first-octant region that is bounded by the coordinate planes, and the additional planes x=1, x+y+z=2.
R is the region common to the three cylinders x2+y2=1, x2+z2=1, y2+z2=1.
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