# 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 .
```
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://studylib.net/doc/15171261/hw-%2319---line-integrals", "fetch_time": 1603582758000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2020-45/segments/1603107885059.50/warc/CC-MAIN-20201024223210-20201025013210-00429.warc.gz", "warc_record_offset": 532800234, "warc_record_length": 12013, "token_count": 542, "char_count": 1268, "score": 3.921875, "int_score": 4, "crawl": "CC-MAIN-2020-45", "snapshot_type": "latest", "language": "en", "language_score": 0.7414312362670898, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00056-of-00064.parquet"}