From zeilberg@math.rutgers.edu Mon Nov 6 19:15:34 2006 Received: (from zeilberg@localhost) by math.rutgers.edu (8.11.7p2+Sun/8.8.8) id kA70EwT13709; Mon, 6 Nov 2006 19:14:58 -0500 (EST) Date: Mon, 6 Nov 2006 19:14:58 -0500 (EST) From: Doron Zeilberger Message-Id: <200611070014.kA70EwT13709@math.rutgers.edu> To: zeilberg@math.rutgers.edu Cc: lpudwell@math.rutgers.edu, scottsch@math.rutgers.edu Subject: MathIsFun; Today's "quiz" Status: RO Content-Length: 1764 Dear Students, Most people got it completely right, but a sizable minority didn't. Please pay attention to the "common mistakes" below, and know to avoid them. -------------Solution to the "quiz" of Nov. 6, 2006-------- Q. Set-up but do not evaluate the iterated integral in spherical coordinates of IntegralSign IntegralSign IntegralSign_E (x^2+y^2+z^2)^100 dV , where E is the region between the hemispheres y=sqrt(1-x^2-z^2) and y=sqrt(4-x^2-z^2). Sol. Since y is positive (sqrt BY DEFINITION is positive), it means that theta ranges from 0 to pi. phi Of course has the full range (0 to pi) and rho goes from the smaller radius (1) to the bigger radius (2). So, E, is: E={(rho,theta,phi)| 1<=rho<=2, 0<=theta<=pi, 0<=phi<=pi} Also x^2+y^2+z^2 is rho^2 (I hope you agree!) and dV=rho^2 sin(phi) d rho d theta d phi So our integral is IntegralSign_{0}^(pi) IntegralSign_{0}^{pi} IntegralSign_ {1}^{2} (rho^2)^100 rho^2 sin (phi) d rho d theta d phi = IntegralSign_{0}^(pi) IntegralSign_{0}^{pi} IntegralSign_ {1}^{2} rho^202 sin (phi) d rho d theta d phi That is the answer! I didn't ask you to evaluate it. ---------------------------------------- Common Mistakes: 1. replaced dV by d rho d theta d phi INSTEAD of rho^2 sin phi d rho d theta d phi 2. Messed up the simple algebra of powers. Some people did : (rho^2)^100 (rho)^2= (rho)^400 ! The correct way, of course is, (rho^2)^100 (rho)^2= (rho)^200 (rho)^2= (rho)^202 3. Some people left (rho)^2 (rho)^2 without simplifying it. While technically it is correct (I didn't ask you to evaluate it) and it is better to leave it like that than mess up like above, you should be so fluent in algebra that (rho^2)^100 (rho)^2= (rho)^200 (rho)^2= (rho)^202 should be in your blood.