Exercises
- Evaluate the triple integral \(\iiint_E xyz^2 \,\mathrm{d}V\) where \(E\) is the rectangular prism \(E = [0,1]\times[-1,2]\times[0,3]\,.\)
- Evaluate \(\iiint_E z \,\mathrm{d}V\) where \(E\) is the solid in the first octant bounded by the surface \(z = 12xy\) and the planes \(y=x\) and \(x=1.\)
- Evaluate \(\iiint_E \sqrt{x^2+z^2} \,\mathrm{d}V\) where \(E\) is the solid bounded by the paraboloid \(y = x^2+z^2\) and the plane \(y=4.\)
- Calculate the volume of the tetrahedron bound by the planes \(x+2y+z=2\) and \(x=2y\) and \(x=0\) and \(z=0.\)
- Sketch the solid over which the iterated integral \(\int_0^{1}\int_0^{x^2}\int_0^{y} \,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x\) 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 \(\mathrm{d}y\,\mathrm{d}x\,\mathrm{d}z \) such that the same volume is computed.
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For each of the following iterated integrals, sketch the solid whose volume is computed by the integral, and write down all the other integrals with the differentials \(\mathrm{d}x\) and \(\mathrm{d}z\) and \(\mathrm{d}y\) permuted that compute the same volume.
\( \displaystyle \int\limits_{0}^{1}\int\limits_{\sqrt{x}}^{1}\int\limits_{0}^{1-y} \,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x\)\( \displaystyle \int\limits_{0}^{1}\int\limits_{0}^{1-x^2}\int\limits_{0}^{1-x} \,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}x\)\( \displaystyle \int\limits_{0}^{1}\int\limits_{y}^{1}\int\limits_{0}^{y} \,\mathrm{d}z\,\mathrm{d}x\,\mathrm{d}y\)\( \displaystyle \int\limits_{0}^{1}\int\limits_{y}^{1}\int\limits_{0}^{z} \,\mathrm{d}x\,\mathrm{d}z\,\mathrm{d}y\)\( \displaystyle \int\limits_0^{2}\int\limits_0^{2-y}\int\limits_0^{4-y^2} \,\mathrm{d}x\,\mathrm{d}z\,\mathrm{d}y\) - 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 \(x=1.\)
- A solid \(E\) 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 \(E?\)
- Evaluate \(\iiint_E x^2 \,\mathrm{d}V\) for the solid \(E\) that lies under the paraboloid \({z=4-x^2-y^2}\) and above the \(xy\)-plane by first expressing the integral in cylindrical coordinates.
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Evaluate this iterated integral by first expressing it in spherical coordinates.
\(\displaystyle \int\limits_{-2}^{2} \int\limits_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int\limits_{\sqrt{x^2+y^2}}^{2} \Big(x^2 + y^2\Big) \,\mathrm{d}z \,\mathrm{d}y \,\mathrm{d}x \) - Calculate the volume of the solid \(S\) that lies above the cone \(z = \sqrt{x^2+y^2}\) but below the sphere \(x^2+y^2+z^2=z.\)
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For the unit ball \(E = \big\{(x,y,z) \mid x^2+y^2+z^2 \leq 1 \big\}\) evaluate
\(\displaystyle \iiint_E \mathrm{e}^{\bigl(x^2+y^2+z^2\bigr)^{3/2}} \,\mathrm{d}V\,. \) - Evaluate \(\iiint_E xyz \,\mathrm{d}V\) over the tetrahedron \(E\) with vertices located at \((0,0,0)\) and \((2,-2,0)\) and \((0,5,0)\) and \((0,0,3).\)
- Evaluate \(\iiint_E \cos(y) \,\mathrm{d}V\) for the solid \(E\) beneath the plane \(x=z\) and above the triangular region with vertices \((0,0,0),\) \((\pi,0,0),\) and \((0,\pi,0).\)
- Evaluate \(\iiint_E xy \,\mathrm{d}V\) for the solid \(E\) bound by the parabolic cylinders \({x=y^2}\) and \({y=x^2}\) and the planes \(z=0\) and \(z=x+y.\)
- TK more examples, and mix them all up tetrahedra
- Active Calculus
- A transformation \(\mathbf{R}^2 \to \mathbf{R}^2\) is defined by the equations \({x=u^2-v^2}\) and \({y=2uv.}\) Sketch a picture of the image of the unit square \({(u,v) \in [0,1]\times[0,1]}\) under this transformation.
- Use the change of coordinates \({x=u^2-v^2}\) and \({y=2uv}\) to evaluate the integral \(\iint_R y \,\mathrm{d}A\) over the region \(R\) bounded by the \(x\)-axis and the parabolas \(y^2=4-4x\) and \(y^2=4+4x\) for \(y \geq 0.\)
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Evaluate the following integral over the trapezoidal region \(R\) with vertices \((1,0)\) and \((2,0)\) and \((0,-2)\) and \((0,-1).\)
\(\displaystyle \iint_R \mathrm{e}^{(x+y)/(x-y)}\,\mathrm{d}A \) -
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
- Imagine a general tetrahedron in space with vertices located at the points with coordinates \(\bigl(x_1, y_1, z_1\bigr)\) and \(\bigl(x_2, y_2, z_2\bigr)\) and \(\bigl(x_3, y_3, z_3\bigr)\) and \(\bigl(x_4, y_4, z_4\bigr).\) Devise a formula for the coordinates of the centroid of this tetrahedron.
- James Stewart The double integral \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xy} \,\mathrm{d}x\,\mathrm{d}y \] is an improper integral and could be defined as the limit of double integrals over the rectangle \([0,t] \times [0,t]\) as \(t\to 1^{-}.\) But if we expand the integrand as a geometric series, we can express the integral as the sum of an infinite series. Show that \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xy} \,\mathrm{d}x\,\mathrm{d}y = \sum_{n=1}^{\infty} \frac{1}{n^2} \]
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James Stewart Leonhard Euler was able to find the exact sum of the series of the reciprocals of square integers. In 1736 he proved that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\] Prove this fact by evaluating the double integral in the previous problem. Start by making the change of variables \[ x \to \frac{u-v}{\sqrt{2}} \qquad y \to \frac{u+v}{\sqrt{2}} \] This gives a rotation about the origin through the angle \(\pi/4.\) You will need to sketch the corresponding region in the \(uv\)-plane. Hint: If, in evaluating the integral, you encounter either of the expressions \(\bigl(1-\sin(\theta)\bigr)/\cos(\theta)\) or \(\cos(\theta)/\bigl(1+\sin(\theta)\bigr),\) you might like to use the identity \(\cos(\theta) = \sin\bigl((\pi/2)—\theta\bigr)\) and the corresponding identity for \(\sin(\theta)\).
- 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 — and it would be a really big deal if someone did find any such expression.