3 Calculus III

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• 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. Comment on that. [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 $z$ around the $z$-axis. 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.