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. 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 with the positive $x$-axis, and the acceleration vector, including its magnitude and angle with the positive $x$-axis?
2. 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]
3. 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 $z$-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 pictures to make your points.[1, Triple Integrals]
4. Try to do the previous question using Cartesian coordinates $x$, $y$ and $z$ instead of cylindrical ones. Work it out at least as far as a single-variable integral, and find the relevant parts in the Math handbook to find its anti-derivative. Use pictures to make your points.
5. 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.