Trivium
Given a solid \(E\) in \(\mathbf{R}^3\) with a boundary that can be described analytically either in rectangular, cylindrical, or spherical coordinates, and a function \(f\colon E \to \mathbf{R},\) compute the value of \[ \iiint\limits_E f \,\mathrm{d}V\,. \]
Given an iterated integral \(\iiint_E f \,\mathrm{d}V\) of a function \(f\colon E \to \mathbf{R}\) on a region \(E \subset \mathbf{R}^3\) and a \(C^1,\) one-to-one transformation \(T\) of \(\mathbf{R}^3,\) compute the Jacobian of the transformation \(\left|\operatorname{J}_T\right|\) and write down the equivalent iterated integral over the pre-transformed domain, \[ \iiint\limits_{T^{-1}(E)} \big(f \circ T\big) \,\bigl|\operatorname{J}_{T}\bigr| \,\mathrm{d}V\,. \]
Exercises
- Evaluate the triple integral \(\iiint_S xyz^2 \,\mathrm{d}V\) where \(S\) is the rectangular prism \(S = [0,1]\times[-1,2]\times[0,3]\,.\)
- 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 \(x=1.\)
- 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 \(y=4.\)
- 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 \(\mathrm{d}y\,\mathrm{d}x\,\mathrm{d}z. \)
- Calculate the volume of the tetrahedron bound by the planes \(x+2y+z=2\) and \(x=2y\) and \(x=0\) and \(z=0.\)
- 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 \(x=1.\)
- Evaluate \(\iiint_S x^2 \,\mathrm{d}V\) for the solid \(S\) that lies under the paraboloid \(z=4-x^2-y^2\) and above the \(xy\)-plane.
- A solid \(S\) lies within the cylinder \(x^2+y^2=1\) on the positive \(y\) side of the \(xz\)-plane, below the plane \(z=4,\) and above the paraboloid \(z=1-x^2-y^2.\) The density at any point in the solid is three times its distance to the \(z\)-axis. What is the mass of \(S?\)
-
Evaluate this iterated integral:
\(\displaystyle \int\limits_{-2}^{2} \int\limits_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int\limits_{\sqrt{x^2+y^2}}^{2} \Big(x^2 + y^2\Big) \,\mathrm{d}z \,\mathrm{d}y \,\mathrm{d}x \)
-
For the unit ball \(S = \big\{(x,y,z) \mid x^2+y^2+z^2 \leq 1 \big\}\)
evaluate
\(\displaystyle \iiint_S \mathrm{e}^{(x^2+y^2+z^2)^{3/2}} \,\mathrm{d}V \)
- Calculate the volume of the solid \(S\) that lies above the cone \(z = \sqrt{x^2+y^2}\) but below the sphere \(x^2+y^2+z^2=z.\)
-
Consider the following change of coordinates
\((\rho, \theta, \phi) \to (x,y,z)\)
and calculate the corresponding Jacobian.
\(\displaystyle x=\rho\sin(\phi)\cos(\theta)\)\(\displaystyle y=\rho\sin(\phi)\sin(\theta)\)\(\displaystyle z=\rho\cos(\phi)\)
Problems & Challenges
-
Consider the tetrahedron in \(\mathbf{R}^3\) with vertices
\((0,0,0)\) and \((0,0,1)\) and \(\big(0,\sqrt{3},-1\big)\) and \(\big(3,\sqrt{3},-2\big)\).
- Write down three integrals — one each based on rectangular, cylindrical, and spherical coordinates — that express the volume of the tetrahedron. Then evaluate one of them to calculate the volume.
- What are the coordinates of the centroid of this tetrahedron?
- What is the surface area of this tetrahedron?
- There exists a change of variables, a transformation of space, that maps the vertices of this tetrahedron to the points \((0,0,0)\) and \((0,0,1)\) and \((0,1,0)\) and \((1,0,0)\). What is this transformation, and what is its Jacobian? Use the transformation and Jacobian to compute the volume of the original tetrahedron.
- James Stewart Show that \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xyz} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z = \sum_{n=1}^{\infty} \frac{1}{n^3} \] So far, no one has been able to express the value of this integral/sum exactly in terms of already-known constants.