![\"\"](\"/userfiles/image/steps_images/step1.JPG\")
(a)
\The parametric equations are ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=49797abdc916390c3f0ef7dc3a30bfda.png\")
and ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=eb5153a804fc84252620eba803c46dd5.png\")
.
Find the rectangular equation.
\Consider ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=49797abdc916390c3f0ef7dc3a30bfda.png\")
.
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=978d5a5682caf730a8e293b978336274.png\")
Squaring on each side.
\![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=80df5927fb3ffb8f79cd1b70792a0a2b.png\")
.
Consider ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=eb5153a804fc84252620eba803c46dd5.png\")
.
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=661b2810d46c7a67b8398b8052e8b081.png\")
Squaring on each side.
\![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=37a4161219fa6b97bc11ea57b8a76f4c.png\")
.
Trigonometric identity: ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=c010bb9f2bd1c342d7c90990a1511122.png\")
.
Substitute ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=80df5927fb3ffb8f79cd1b70792a0a2b.png\")
and ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=37a4161219fa6b97bc11ea57b8a76f4c.png\")
.
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=8478c5a9edebeab363f5a73e18d66c13.png\")
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=4571410cae3acc4d380e2cb7b70d4ff7.png\")
The rectangular equation is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=4571410cae3acc4d380e2cb7b70d4ff7.png\")
.
Graph:
\Graph the parametric equations ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=df113528e406ef4bd665248fbd53ef23.png\")
and ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=068f978c225571694e373b88552601b7.png\")
.
![\"\"](\"/userfiles/48-718-34a.gif\")
.
Observe the graph:
\The domain of the graph is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=9904bb7522639698aeb17b88dc88ebcb.png\")
.
It is circle with Anticlockwise orientation and curve is smooth and continuous on the interval.
\The range is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=b3e661da5347f1a528a27e3b4e99c595.png\")
.
![\"\"](\"/userfiles/image/steps_images/step2.JPG\")
(b)
\The parametric equations are ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=70133ef35929e833c9db32e87214b205.png\")
and ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=6ca5f6437a5dafbe7364efed77eddf08.png\")
.
Graph:
\Graph the parametric equations ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=70133ef35929e833c9db32e87214b205.png\")
and ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=6ca5f6437a5dafbe7364efed77eddf08.png\")
.
![\"\"](\"/userfiles/48-718-34b.gif\")
.
Observe the graph:
\The domain of the graph is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=f72cbbada94fc8389d946760472a0332.png\")
.
It is semicircle with clockwise orientation and curve is smooth and continuous on the interval.
\The range is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=b3e661da5347f1a528a27e3b4e99c595.png\")
.
![\"\"](\"/userfiles/image/steps_images/step3.JPG\")
(c)
\The parametric equations are ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=b1a56abf2ba91d34b0d2a4f1bbf415ec.png\")
and ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=813bee2496a68603a774db7daa69207e.png\")
.
Graph:
\Graph the parametric equations ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=b1a56abf2ba91d34b0d2a4f1bbf415ec.png\")
and ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=813bee2496a68603a774db7daa69207e.png\")
.
![\"\"](\"/userfiles/48-718-34c.gif\")
.
Observe the graph:
\The domain of the graph is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=f72cbbada94fc8389d946760472a0332.png\")
.
It is quater circle with clockwise orientation and curve is smooth and continuous on the interval.
\The range is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=51141e31a6271e0918080e8c62ecc431.png\")
.
![\"\"](\"/userfiles/image/steps_images/step4.JPG\")
(d)
\The parametric equations are ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=762303cdfc6874af11365fa1aa951044.png\")
and ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=1cb0d4285097b3bf4f837ecacc0d59e2.png\")
.
Graph:
\Graph the parametric equations ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=762303cdfc6874af11365fa1aa951044.png\")
and ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=1cb0d4285097b3bf4f837ecacc0d59e2.png\")
.
![\"\"](\"/userfiles/48-718-34d.gif\")
.
Observe the graph:
\The domain of the graph is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=9904bb7522639698aeb17b88dc88ebcb.png\")
.
It is semi circle with clockwise orientation and curve is smooth and continuous on the interval.
\The range is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=51141e31a6271e0918080e8c62ecc431.png\")
.
Observe the set of graphs of parametric equations:
\The parametric equations from (a)-(d) are similar and their rectangular equation is .
They differ in orientation and some are domain restricted.
\![\"\"](\"/userfiles/image/steps_images/solution.JPG\")
The parametric equations from (a)-(d) are similar and their rectangular equation is .
They differ in orientation and some are domain restricted.
\(a)
\The domain of the graph is .
It is semicircle with clockwise orientation.
\The range is .
(b)
\The domain of the graph is .
It is semicircle with clockwise orientation.
\The range is .
(c)
\The domain of the graph is .
It is quater circle with clockwise orientation.
\The range is .
(d)
\The domain of the graph is .
It is semi circle with clockwise orientation.
\The range is .