MATH 241 Lecture Notes - Lecture 19: Polar Coordinate System

• School: University of Maryland
• Department: Mathematics
• Course: MATH 241
• Professor: Christian Rosendal

Document Summary

Math241 lecture 19 double and triple integrals (continued) Example: find the volume below (cid:1858)(cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:884)+(cid:1876)+(cid:1877) and above the region which is bound by the lines (cid:1877)=(cid:1876),(cid:1877)=(cid:884)(cid:1876),(cid:1877)=(cid:884). How about as horizontally simple? (cid:882) (cid:1876) (cid:883), (cid:883) (cid:1876) (cid:884), (cid:1872)(cid:1871), (cid:1872)(cid:1871), (cid:1877)=(cid:884)(cid:1876) (cid:1877)=(cid:884) 2 y y left y x= x y right d = So we can do (2 x y dx dy. Introduction: often the case that a region is not nice in vertically simple or horizontally simple form. Descriptions of in polar: a region in polar is described as follows. 3 x (cid:2009)=(cid:882), (cid:1866)(cid:1866)(cid:1857)(cid:1870) (cid:1858)(cid:1873)(cid:1866)(cid:1855)(cid:1872)(cid:1867)(cid:1866):(cid:1870)=(cid:882) (cid:1868)=(cid:886), (cid:1867)(cid:1873)(cid:1872)(cid:1857)(cid:1870) (cid:1858)(cid:1873)(cid:1866)(cid:1855)(cid:1872)(cid:1867)(cid:1866):(cid:1870)=(cid:885) is the region outside (cid:1870)=(cid:883) is the region inside (cid:1870)=(cid:884)cos. Note: do not use (cid:2873)(cid:2871) because it misses the region. Suppose is a region described in polar with (cid:2009),(cid:2010), inner , outer. (cid:1858)(cid:4666)(cid:1870)cos,(cid:1870)sin(cid:4667) (cid:3081) (cid:3042)(cid:3048)(cid:3047) (cid:1856)= (cid:3041)(cid:3041) (cid:3080) = first picture (cid:1856)= (cid:1870)(cid:2870) (cid:1876)(cid:2870)+(cid:1877)(cid:2870)=(cid:1870)(cid:2870) (cid:2872)(cid:2868) (cid:2872)(cid:2868) (cid:1870)(cid:1856)(cid:1870)(cid:1856)= (cid:1870)(cid:2870)(cid:2871) (cid:2871) = second picture (cid:2871) (cid:2871) (cid:2870)cos (cid:1856)= (cid:2869)