Question Q1 Use spherical coordinates to evaluate the triple integral SI/ (0 – 3) dxdydz, - where D is the solid region defined by the inequalities 22 + y2 +254, x > 0 and y 2 0.
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Transcribed Image Text: Q1 Use spherical coordinates to evaluate the triple integral SI/ (0 – 3) dxdydz, - where D is the solid region defined by the inequalities 22 + y2 +254, x > 0 and y 2 0.
More Transcribed Image Text: Q1 Use spherical coordinates to evaluate the triple integral SI/ (0 – 3) dxdydz, - where D is the solid region defined by the inequalities 22 + y2 +254, x > 0 and y 2 0.
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(1) ∭_(D)(y-z)dxdydz.{:[x^(2)+y^(2)+z^(2) <= 4],[x >= 0quad y >= 0.]:}given region is sphere in ist quadrant.{:[quadx^(2)+y^(2)+z^(2)=4=e^(2)],[:.e,=2],[0 <= e <= 2],[x,ee sin phi cos theta],[y,e sin phi sin theta],[z,=e cos phi],[dxdydz=e^(2)sin phi ded theta d phi],[0 <= phi <= pi//2],[0 <= theta <= pi//2],[0 <= e <= 2.]:}{:[int_(0)^(pi//2)int_(0)^(pi//2)int_(0)^(2)(e sin phi sin theta-e cos phi)e^(2)sin phi],[ded theta d phi],[=>int_(0)^(pi//2)int_(0)^(pi//2)int_(0)^(2)(e^(3)d)(sin^(2)phi sin theta-sin phi cos phi)],[ ... See the full answer