Question

Parametrize the surface of a sphere of radius R center at the origin using Knowledge of surface integral Use the parametrization to set up the integral to Compute the surface area of a sphere of radius R. and evaluate the integral

Answer 1

solution we use the spherical coordinate P(r, 0,0) system as in pictured. y We take angle anticlockwise sense. so the parametrization of sphere with be of radius R will be x=Roso sind y= Rsino sind. and. OcØLT. z = Raso. osocza (Aus) In general. if we have the coordinate system. x=ro Lose sino Y = rsana sino za plno. then elementary surface area. dA=lal do do. where [J/- determinent of the Jacobitan transformation matrix Y2 Er Ayo - zo no 90 Yo zo rsino cup arsiup . abe sing sino sino Cose arsinosing a loosing o 8 GBOGBO = Crad (-resinto sin plago.-82650 sin play) -rsino (8320 sino + rsino sino) - preso sin Ø 430 (1) - Josina postura (1) -p2 sino dA= -p2 sino do do. a. dA=-R² sino do do. surface area of the sphere of radius | 27 _R2 sing do do 0=0=0 = {2ARY {-co38] Bar - 20R (14) 4 TRY tve sense. In our lase. VaR. 0 O A 2 E2ARY ( sind do dan [:0 in tanken canticlockwise)]

solution we use the spherical coordinate P(r, 0,0) system as in pictured. y We take angle anticlockwise sense. so the parametrization of sphere with be of radius R will be x=Roso sind y= Rsino sind. and. OcØLT. z = Raso. osocza (Aus) In general. if we have the coordinate system. x=ro Lose sino Y = rsana sino za plno. then elementary surface area. dA=lal do do. where [J/- determinent of the Jacobitan transformation matrix Y2 Er Ayo - zo no 90 Yo zo rsino cup arsiup . abe sing sino sino Cose arsinosing a loosing o 8 GBOGBO = Crad (-resinto sin plago.-82650 sin play) -rsino (8320 sino + rsino sino) - preso sin Ø 430 (1) - Josina postura (1) -p2 sino dA= -p2 sino do do. a. dA=-R² sino do do. surface area of the sphere of radius | 27 _R2 sing do do 0=0=0 = {2ARY {-co38] Bar - 20R (14) 4 TRY tve sense. In our lase. VaR. 0 O A 2 E2ARY ( sind do dan [:0 in tanken canticlockwise)]