Math 230 Calculus IISummer 2004


7 problems, 10 points each


1. Show enough work to justify your answers

2. Don't use calculators for integration.

3. Don't leave until you are sure you've solved all the problems.

  1. Let MATH (a) Use the graph of $y=f(x)$ below to estimate $L_{4},M_{4},R_{4}.$ (b) Compare them to $I$.

  2. Evaluate MATH

  3. Find the area of the surface of revolution around the $x$-axis obtained from MATH

  4. Find the centroid of the region bounded by the curves $y=4-x^{2},y=x+2$.

  5. Describe the motion of a particle with position $(x,y),$ where $x=2+t\cos t,$ $y=1+t\sin t,$ as $t$ varies within $[0,\infty )$ .

  6. Suppose the parametric curve is given by MATH Set up, but do not evaluate, the integrals that represent (a) the arc-length of the curve$,$ (b) the area of the surface obtained by rotating the curve about the $x$-axis.

  7. Plot the curve $r=2\cos (3\theta )$. For 5 extra points find the line(s) through the origin tangent to the curve.


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