3 Calculus III

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Variable $z$ is given in terms of the measurable variables $f$ and $g$ as
   
   $$\frac1z = \frac1f + \frac1g$$
   
   
   The values of $f$ and $g$ and their uncertainties are:
   
   $$f = 4 \pm 0.01 \qquad g = 8 \pm 0.02$$
   
   
   What are the maximum relative and absolute errors in the computed $z$? Are you stunned by the value of the relative error in $z$? Explain why not. [1, Total Differential]
2. A particle moves in the first quadrant along the parabola $y^2=12x$. The x-component of velocity is $v_x=15$. At the point (3,6), what are the velocity vector, including its magnitude and angle $\tau$ with the positive $x$-axis, and the acceleration vector, including its magnitude and angle $\phi$ with the positive $x$-axis?
3. Find $I_x$ for the area between the curves
   
   $$y = x \qquad y = 4x -x^2$$
   
   
   Exact answers only, please. Since the integrand $y^2$ does not depend on $x$, it would seem logical to integrate $x$ first. Discuss that in detail. [1, Centroids and Moments of Inertia]
4. Find the volume of the region bounded by
   
   $$z=0 \qquad x^2 + y^2 = 4x \qquad x^2 + y^2 = 4 z$$
   
   
   Use cylindrical coordinates $r$, $\theta$ (or $\phi$ if you want), and $z$ (the normal ones around the given $z$-axis). List the limits if (a) you do $z$ first, (b) you do r first, and (c) you do $\theta$ first. To do the latter two cases, make a picture of the cross-section of the region for a fixed value of $z$ like $z=\frac14$ and show the $r$ and $\theta$ integration lines. What variable is obviously the one to integrate first? For the second integration, discuss each possibility and explain which is the best choice. Use neat pictures to make your points.[1, Triple Integrals]
5. Try to do the previous question's integral using Cartesian coordinates $x$, $y$ and $z$ instead of cylindrical ones. Work it out at least as far as the final single-variable integral, and find the relevant parts in the Math handbook to find its anti-derivative. Putting in the numbers can be skipped. Use neat pictures to make your points.
6. Evaluate the integral
   
   $$\int_1^2 x \sqrt[3]{x^5+2x^2-1} {\rm d} x$$
   
   
   to 6 digits accurate using 5 function values spaced 0.25 apart. Use both the trapezium rule for four strips and the Simson rule for two double strips. Compare results to the exact value 3.571639.