- #1

dyn

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- Homework Statement
- Consider the volume defined by the surface x^2 + (y-2)^2 +z^2 = 4 where x , y , z > 0 and the x=0 , y=0 , z=0 planes

(a) sketch this volume

(b) write down the definite integral which defines the volume in Cartesian coordinates

(c) integrate the function 1/ ( x^2 + (y-2)^2 +z^2 ) over this volume by making an appropriate coordinate transformation

- Relevant Equations
- Equation of sphere x^2+y^2+z^2 = constant , volume element in spherical polars is

dV = r^2 sin (theta) dr d(theta) d(phi)

(a) i sketched a quarter of a sphere centred at x=0 , y=2 , z=0

(b ) ∫ ∫ √ (4-x

(c ) i converted to spherical polars and took the integrand as 1/r

This leads to the triple integral of sinθ with limits 0< r <2 , 0 <θ <π/2 , 0 < ∅ < π

This does give the correct answer of 2π but i am not sure if i can just take the integrand as 1/r

How have i done with all 3 parts ?

Thanks

(b ) ∫ ∫ √ (4-x

^{2}- (y-2)^{2}) dx dy with limits 0 < x < 2 and 0 < y <4(c ) i converted to spherical polars and took the integrand as 1/r

^{2}. the volume element is r^{2}sinθ drdθd∅This leads to the triple integral of sinθ with limits 0< r <2 , 0 <θ <π/2 , 0 < ∅ < π

This does give the correct answer of 2π but i am not sure if i can just take the integrand as 1/r

^{2}because the sphere is displaced from the origin. But it does give the correct answer !How have i done with all 3 parts ?

Thanks