Transcription of Solution to Problem Set #9 - utoledo.edu
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Solution to Problem Set #9 - utoledo.edu
Solution to Problem Set # the area of the following surface.(a)(15 pts) The part of the paraboloidz= 9 x2 y2that lies abovethex yplane. 4 2024x 4 2024y 4 part of the paraboloidz= 9 x2 y2that lies abovethex yplane must satisfyz= 9 x2 y2 0. Thusx2+y2 9. Wehavez=f(x, y) = 9 x2 y2,fx= 2x,fy= 2yand 1 +f2x+f2y= 1 + ( 2x)2+ ( 2y)2= 1 + 4x2+ regionE={(x, y)|x2+y2 9}is{(r, )|0 r 3,0 2 }inpolar coordinates. Hence the area of surface is E 1 +f2x+f2ydxdy= 2 0 30 1 +r2rdrd = 2 013(1 +r2)32|30d =(13(10)32 13) 2 =2 3((10)32 1). (b)(15 pts)The part of the spherex2+y2+z2= 4that lies above theplanez= 1. 4 2024x 4 2024y 4 spherex2+y2+z2= 4can be written asz= 4 x2 part of the spherex2+y2+z2= 4that lies above the planez= 1must satisfy1 z= 4 x2 y2. Thusx2+y2 3.
Solution to Problem Set #9 1. Find the area of the following surface. (a) (15 pts) The part of the paraboloid z = 9 ¡ x2 ¡ y2 that lies above the x¡y plane. ±4 ±2 0 2 4 x ±4 ±2 0 2 4 y ±4 ±2 0 2 4 Solution. The part of the paraboloid z = 9¡x2 ¡y2 that lies above the x¡y plane must satisfy z = 9¡x2 ¡y2 ‚ 0. Thus x2 +y2 • 9. We
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