Practice Exam - Math 13 - Feb 99'

1. Find the volume inside the sphere x²+y²+z²=4a² and outside the cylinder x²+y²=2ax.

2. Using polar coordinates, evaluate

$$\int_{-a}^a\int_0^{\sqrt{a^2-y^2}}(x^2+y^2)^{\frac{3}{2}}dx\,dy$$

3. Use Lagrange multipliers to prove that the triangle with maximal area that has a given perimeter P is equilateral.[HINT: Use Heron's formula for the area : $A=\sqrt{s(s-x)(s-y)(s-z)}$ where $s=\frac{P}{2}$ and x,y and z are the lengths of the sides].

4. Let S be the surface

$$S=\{(x,y,z)\, \vert\, x^2+y^2\leq4 ; z=(x-1)^2+(y-1)^2+1 \}$$

Find the point of S which is the closest to the xy plane.

5. Find the tangential and normal components of the acceleration of ${\bf r}(t)=t{\bf i}+\cos^2(t) {\bf j}+\sin^2(t){\bf k}$. Compute the curvature of the trajectory at time t=0.