Triple Integrals in Rectangular Coordinates
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Stewart
Evaluate the triple integral \(\iiint_S xyz^2 \,\mathrm{d}V\)
where \(S\) is the rectangular prism
\(S = [0,1]\times[-1,2]\times[0,3]\,.\)
-
Stewart
Evaluate \(\iiint_S z \,\mathrm{d}V\)
where \(S\) is the solid in the first octant
bounded by the surface \(z = 12xy\)
and the planes \(y=x\) and \(x=1.\)
-
Stewart
Evaluate \(\iiint_S \sqrt{x^2+z^2} \,\mathrm{d}V\)
where \(S\) is the solid bounded by
the paraboloid \(y = x^2+z^2\) and the plane \(y=4.\)
-
Stewart
Sketch the solid over which the iterated integral
\(\int_0^{1}\int_0^{x^2}\int_0^{y} f(x,y,z) \,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x\)
is concerned, and then rewrite the iterated integral
with the order of the variables permuted:
first with respect to \(\mathrm{d}x\,\mathrm{d}z\,\mathrm{d}y,\)
then with respect to \(\mathrm{d}y\,\mathrm{d}x\,\mathrm{d}z. \)
-
Stewart
Calculate the volume of the tetrahedron bound by the planes
\(x+2y+z=2\) and \(x=2y\) and \(x=0\) and \(z=0.\)
-
Stewart
Calculate the coordinates of the center of mass
of the solid with constant density that is bounded by
the parabolic cylinder \(x=y^2\)
and the planes \(x=z\) and \(z=0\) and \(x=1.\)
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Review
What are the cylindrical coordinates \((r,\theta,z)\)
of a point in \(\mathbf{R}^3\)
with rectangular coordinates \((x,y,z)=(8,-4,3)?\)
-
Review
What are the rectangular coordinates \((x,y,z)\)
of a point in \(\mathbf{R}^3\)
with cylindrical coordinates \((r,\theta,z)=(7,\pi/6,2)?\)