Math 206A Quiz 06 page 1 Friday 11/19/2010 Name
1. Consider the path parameterized by f(t) = (t3/3−t, sin t) for t∈[−π , π].
1A) Make a nice sketch of this path in the window [−8,8] ×[−1,1]. Use Tstep=0.1 Make an excellent facsimile of your
calculator’s drawing on the axes provided. The Xscl and Yscl values are 1 and 0.5 respectively.
1B) Set up (but don’t try to evaluate) the integral which represents the arc length of this path.
1C) Use your calculator’s numeric integration program to find to two places after the decimal point, the length of this
path. (NB: you may need to switch your calculator from Par back to Func after drawing the path for part (1A) before you
can use the built-in Rf(x)dx feature).
2) A “fence” is built over the path parameterized by f(t) = (t, t2−9) for t∈[0,3]. The height of the fence over any point
(x, y) on the path is 2x+ 3y+ 27. Set up (but don’t evaluate) the integral that represents the area of one side of the fence.
3) The path parameterized by f(t) = (t, t2−9) for t∈[0,3] runs through the vector field F(x, y) = −−−→
(y, x). If this field
represents a force, find the work done by the force on an object moving on that path with the given parameterization. Show
all your work, from the integral you need to set up, to its evaluation.