1. Find the length of the curve for . Hint: Use and parameterization.
2. Show that the arc length is independent of the functions below that parameterize the curve. Compute the length of the semicircle of radius 1 and center at (0,0) by using the two different parametric equations of the circle given below and show that the arc length is the same. This argument suggests that ark length is independent of parameterization; though, it is not a proof of it as we would need to show that this holds for all parameterizations.
a) where
b) where
3. Find the length of the Arc of St. Louis, if the equation used in construction the arc is where
4. Find the center of mass of the trapezoid with constant density 1 and with vertices at (0, 0), (c, 0), (c, b), and (0, a) where a, b, and c are constants. Draw the trapezoid on the plane provided and show it is the intersection of the line connecting the midpoint of the parallel sides and the line connecting the extended parallel sides.
5. Find the center of mass of the region bounded by the graphs of and . Assume the density is constant and is equals to 1. Make a sketch of the region and identify its center of mass.
6. An ornamental light bulb is designed by revolving the graph of about the x-axis where x and y are measured in feet. Find S, the surface area of the bulb.
7. Find the surface area generated by revolving the curve f: [1,5] where around the x-axis.
8. Use the theorem of Pappus to find the volume generated by revolving about the line the triangular region bounded by the coordinate axis and the line .
9. Curves represented by the graphs of the equation are called astroids because of their shapes which look like stars.
a) Show that the graph represented by this equation is symmetric with respect to the y-axis as well as with respect to the x-axis.
b) Find the length of the astroid. Hint : Find the length of half of the first quadrant portion using the function ; x and then multiply by 2 and use a)
c) Find the area of the surface generated by revolving the portion of the astroid represented by the graph above the x-axis. Hint: Revolve the graph in the first quadrant about the x-axis and double it.
10. Gabriel’s horn:
a) Compute volume obtained by revolving around x-axis for
b) Compute surface obtained by revolving around x-axis for
