We are asked to find the volume of the solid bounded by the sphere
=
ρ
6cosφ
and the elliptical
cone
=
+
z
2x2 3y2
In spherical coordinates,
=
, =
, =
,
=
+
+
x
ρsinφcosθ
y
ρsinφsinθ
z
ρcosφ
and ρ
x2 y2 z2
.
Change the
equation of the sphere to Cartesian coordinates:
+
+
=
+
+
x2 y2 z2
6zx2 y2 z2
+
+
=
x2 y2 z2
6z
+
+

+ =
x2 y2 z2 6z 9
9
+
+ 
=
x2 y2 z 32
9
which is the equation of a sphere with radius 3 and center (0,0,3). It's projection onto the xyplane is a
circle of radius 3 and
center at the origin
.
As for the elliptical cone, it's projection onto the xy plane is an ellipse with
center at the origin.
So, the projection on the xyplane bounded by both this circle and ellipse is going to be region that
goes completely around the xy plane. Thus, the bounds of
θ
in spherical coordinates of the 3D solid
will be
0 to 2π
.
The lower bound of
φ
is 0. To find the upper bound, switch the equation of the elliptical cone to
spherical coordinates and simplify:
=
+
ρcosφ
2ρ2sin2φcos2θ 3ρ2sin2φsin2θ
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 Spring '11
 Prof.Ramachandran
 Polar coordinate system

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