Math113 Calculus III Exam3 Practice Problems (Fall2005)
Prepared for students who want a focused review before the third exam of the semester. The problems below reflect the style and difficulty of the 2005 Fall exam, covering multivariable integration, vector calculus, and series.
Triple Integrals
Problem1. Evaluate the integral \[ \iiint_{E} (x^{2}+y^{2})\,dV, \] where \(E\) is the solid bounded by the paraboloid \(z = 4 - x^{2} - y^{2}\) and the plane \(z=0\).
Convert to cylindrical coordinates: \(x = r\cos\theta,\; y = r\sin\theta,\; dV = r\,dr\,d\theta\,dz\). The region is \(0\le r\le 2,\;0\le\theta\le2\pi,\;0\le z\le 4-r^{2}\). The integrand becomes \(r^{2}\). The integral evaluates to \(\displaystyle \int_{0}^{2\pi}\int_{0}^{2}\int_{0}^{4-r^{2}} r^{3}\,dz\,dr\,d\theta = 64\pi/3.\)
Problem2. Compute \[ \iiint_{E} \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}\,dV, \] where \(E\) is the region inside the sphere \(x^{2}+y^{2}+z^{2}=9\) but outside the cone \(\displaystyle z = \sqrt{x^{2}+y^{2}}\).
Spherical coordinates are ideal: \(\rho\) from 0 to 3, \(\phi\) from \(\pi/4\) to \(\pi\), \(\theta\) from 0 to \(2\pi\). The integrand simplifies to \(1/\rho\) and the Jacobian is \(\rho^{2}\sin\phi\). Integral = \(\displaystyle \int_{0}^{2\pi}\int_{\pi/4}^{\pi}\int_{0}^{3} \rho \sin\phi\,d\rho\,d\phi\,d\theta = 27\pi (1-\frac{1}{\sqrt{2}}).\)
Cylindrical and Spherical Coordinates
Problem3. Transform the region bounded by the cylinders \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=4\) and the planes \(z=0\) and \(z= x\) into cylindrical coordinates and write the integral \[ \iiint_{E} z\,dV. \]
In cylindrical form, \(1\le r\le 2\), \(0\le\theta\le2\pi\), \(0\le z\le r\cos\theta\). The integral becomes \(\displaystyle \int_{0}^{2\pi}\int_{1}^{2}\int_{0}^{r\cos\theta} z\, r\,dz\,dr\,d\theta.\) After evaluation the result is \(\displaystyle \pi/4.\)
Problem4. Find the volume of the solid that lies inside the sphere \(\rho = 5\) and above the cone \(\phi = \pi/3\).
Limits: \(\rho\) from 0 to 5, \(\phi\) from 0 to \(\pi/3\), \(\theta\) from 0 to \(2\pi\). Volume = \(\displaystyle \int_{0}^{2\pi}\int_{0}^{\pi/3}\int_{0}^{5}\rho^{2}\sin\phi\,d\rho\,d\phi\,d\theta = \frac{125\pi}{3}\bigl(1-\cos(\pi/3)\bigr)=\frac{125\pi}{6}.\)
Vector Fields & Line Integrals
Problem5. Let \(\mathbf{F}(x,y,z)=\langle y,\;x,\;z^{2}\rangle\). Compute the line integral \(\displaystyle\int_{C}\mathbf{F}\cdot d\mathbf{r}\) where \(C\) is the triangle with vertices \((0,0,0)\), \((1,0,0)\) and \((1,1,1)\) traversed counterclockwise when viewed from above.
Break \(C\) into three segments and evaluate directly, or note that \(\nabla\times\mathbf{F}= \langle 0,0,-2\rangle\) is not zero, so the field is not conservative. Direct calculation gives a total of \( \frac{5}{6}.\)
Problem6. Determine whether \(\mathbf{F}(x,y)=\langle -y/(x^{2}+y^{2}),\;x/(x^{2}+y^{2})\rangle\) is conservative on \(\mathbb{R}^{2}\setminus\{(0,0)\}\) and, if so, find a potential function.
Compute \(\partial P/\partial y = \partial Q/\partial x\); they are equal, but the domain is not simply connected. The field is not conservative; its circulation around the unit circle equals \(2\pi\).
Surface Integrals & Flux
Problem7. Find the flux of \(\mathbf{F}(x,y,z)=\langle x,\,y,\,z\rangle\) outward through the part of the paraboloid \(z=4-x^{2}-y^{2}\) that lies above the plane \(z=1\).
Parameterize the surface as \(\mathbf{r}(r,\theta)=\langle r\cos\theta,\,r\sin\theta,\,4-r^{2}\rangle\) with \(0\le r\le\sqrt{3}\). The outward normal has magnitude \(\sqrt{1+4r^{2}}\). After simplifying, the flux integral reduces to \(\displaystyle \int_{0}^{2\pi}\int_{0}^{\sqrt{3}} (4-r^{2}) r\,dr\,d\theta = 8\pi.\)
Problem8. Use the Divergence Theorem to evaluate \(\displaystyle\iiint_{E}\nabla\!\cdot\!\mathbf{F}\,dV\) for \(\mathbf{F}= \langle yz,\,xz,\,xy\rangle\) over the cube \(0\le x,y,z\le 1\).
\(\nabla\!\cdot\!\mathbf{F}= x+y+z\). The triple integral = \(\displaystyle\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}(x+y+z)\,dx\,dy\,dz = \frac{3}{2}.\) By the Divergence Theorem the same value is the total flux across the six faces.
Series & Convergence
Problem9. Determine whether the series \(\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n}n}{n^{2}+1}\) converges absolutely, conditionally, or diverges.
The absolute series behaves like \(\sum n/n^{2}= \sum 1/n\), which diverges. The alternating series satisfies the Leibniz test because \(\frac{n}{n^{2}+1}\) decreases to 0. Hence the series converges conditionally.
Problem10. Find the radius of convergence for the power series \(\displaystyle\sum_{k=0}^{\infty}\frac{(2k)!}{k!}\,x^{k}\).
Apply the ratio test: \(\displaystyle\lim_{k\to\infty}\frac{(2k+2)!}{(k+1)!}\cdot\frac{k!}{(2k)!}= \lim_{k\to\infty}\frac{(2k+2)(2k+1)}{k+1}= \infty.\) Therefore the radius of convergence is 0; the series converges only at \(x=0.\)
ExamDay Tips for CalculusIII
- Read the problem carefully. Many points are lost by misreading the limits or the orientation of a surface.
- Sketch the region. Even a quick diagram helps set up correct bounds, especially for triples integrals.
- Choose coordinates wisely. If a surface is a sphere or a cone, switch to spherical coordinates; for cylinders, use cylindrical.
- Check for symmetry. Odd functions over symmetric intervals vanish; this can save time.
- When a vector field is conservative, use a potential function. This avoids messy line integrals.
- Use the Divergence and Stokes theorems. They turn surface or line integrals into easier volume or area integrals.
- For series, first test absolute convergence. If it fails, apply alternating or comparison tests.
- Keep an eye on units. In physicsstyle problems, forgetting a factor of \(r\) or \(\sin\phi\) is a common mistake.
- Allocate time. Spend no more than a few minutes on a problem that feels stuck; move on and return later.
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