Math 174 02,

the course of Dr. Mihailovs

Midterm 3

November 20, 1998

  1. Find the area of the surface obtained by rotating the curve $ y=3\sqrt{x}+1/(2\sqrt{x})$ from $ 1$ to $ 4$ about the $ x$-axis.

  2. Find the Cartesian equation of the curve $ x=3\cos t$, $ y=2\sin t$ .

  3. Find the equation of the tangent to the curve $ x=t^2+1$, $ y=2t\sin t$ at $ t=0$ .

  4. Find the area of the region bounded by the curve in Problem 2.

  5. Find the length of Cornu's spiral $ x=\int_0^t\cos\frac{\pi u^2}{2} \; du$ , $ y=\int_0^t\sin\frac{\pi u^2}{2} \; du$ from $ t=0$ to $ t=2$ .

  6. Find the area inside the circle $ r=2$ and outside the cardioid $ r=2-2\sin \theta$ .

  7. Find the eccentricity, identify the conic, give an equation of the directrix and sketch the conic $ r=\frac{3}{2+3\cos\theta}$ .

  8. Find $ \sum_{k=1}^{\infty}\frac{a_k}{k(2k+1)}$ where $ a_k=\sum_{n=0}^{\infty}\frac{1}{(2k)^n}$ .

  9. Test the series $ \sum_{n=2}^{\infty}\frac{1}{n\ln^2n}$ for convergence or divergence.

Copyright © 1998 Aleksandrs Mihailovs