Question Q1 Use spherical coordinates to evaluate the triple integral SIT,(= -2 (3 - y) dadydz, where D is the solid region defined by the inequalities 22 + y2 + x2 < 9, x > 0 and 2 > 0. [12 marks]

4QPJWD The Asker · Algebra

Transcribed Image Text: Q1 Use spherical coordinates to evaluate the triple integral SIT,(= -2 (3 - y) dadydz, where D is the solid region defined by the inequalities 22 + y2 + x2 < 9, x > 0 and 2 > 0. [12 marks]

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Hi, I attached the step by step solution for your questions in very easy and understandable way.please find it. Solutionbiiven thatby using spherical co-ordinates evallaateI=∭_(0)(x-y)dxdydzD:x^(2)+y^(2)+z^(2) <= 9,x >= 0" and "z >= 0We know thatin spherical co-ardinates sys km{[x=r sin theta*cos varphi],[y=r sin theta*sin varphi],[z=r cos theta]:}By using above relation we have to find the limit for r,theta and varphiGiven that{:[x^(2)+y^(2)+z^(2)=9],[(r sin theta*cos varphi)^(2)+(r sin theta*sin varphi)^(2)+(r cos theta)^(2)=9],[r^(2)sin^(2)theta(cos^(2)varphi+sin^(2)varphi)+r^(2)cos^(2)theta=9]:}CS Scanned with CamScanner{:[r^(2)(sin^(2)theta+cos^(2)theta)=9],[r^(2)=9],[r=13]:}x^(2)+y^(2)+z^(2)=9Given z=0x^(2)+y^(2)=9Consider{:[x^(2)+y^(2)=r^(2)sin^(2)theta(sin^(2)varphi+cos^(2)varphi)],[9=3^(2)sin^(2)theta xx1],[sin^(2)theta=1],[sin theta=11],[theta=pi//2quad" consider ondy in "],[" tirst Quaodrant "]:}limit for theta0 <= theta <= (pi)/(2)CS Scanned with CamScannerGiven{:[x^(2)+y^(2)+z^(2)=9],[x >= 0","2 >= 0],[x=0","2=0],[0^(2)+y^(2)+0^(2)=9],[y^(2)=9],[y=+-3]:}we know that{:[sin varphi=(y)/(sqrt(x^(2)+y^(2)))","6" l ... See the full answer