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
-
Remember to replace
h4withspan class="h5". -
TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK
\(\displaystyle y = mx+b\)\(\displaystyle y = mx+b\)
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.