# HW #19 - Line Integrals

```Line Integrals
Suggested Problems: 6.1: 1,3,4,7,9,12,14,15, 25, 28;
1) a) Consider the quarter-ellipse, E, in the first quadrant parameterized by
 x  2cos 
, 0     / 2 . Find the path integral  xy ds .

E
 y  sin 
x2
b) Change the parameterization of this curve to y  1  , 0  x  2 and redo the integral.
4
c) How do the parameterizations from a) and b) compare? How do the results from a) and b)
compare? Explain.
2) a) Let r  r (), a    b, be a curve C given in polar coordinates. Let f(x, y) be a function.
Derive the formula for

C
f ds in terms of .
b) Use your formula to find the arc-length of the cardiod r = 1 + cos .
3) Find parameterizations for the following curves:
a) The line segment from (1, 2,3) to (2, 4, 1)
b) A spring with four coils, axis along the x-axis, radius 2, length along x-axis is 1, spiraling
clockwise as x increases.
c) A spiral drawn on the cone z = r (choose a convenient one).

4) Let x (t )  (t , t 2 , t 3 ), 0  t  2. Let F(x, y, z) = xyi + yzj + zxk. Evaluate
5) Let the path x (t ) be given by (t, t2, 1), 1  t  3. Evaluate
x
c
2

F  ds .
dx  xy dy  xyz dz .
6) Let C be the directed line segment from (2, 1, 3) to (3, 0, 1). Evaluate

C
x2 dx  xy dy  dz .
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