Question

Use spherical coordinates to evaluate the following integral, ∫ ∫ ∫ y2z dV, E where E lies above the cone φ = π 4 and below the sphere ρ = 9

Answer #1

Use spherical coordinates.
Evaluate
xyz
dV
E
,
where E lies between the spheres ρ = 2 and
ρ = 5 and above the cone ϕ = π/3.

Use spherical coordinates. y^2z^2dV, where E lies below the cone
ϕ = π/3 and above the sphere ρ = 1.

Use spherical coordinates.
Evaluate
(x2 + y2) dV
E
,
where E lies between the spheres
x2 + y2 + z2 = 9 and
x2 + y2 + z2 = 16

7. Given The triple integral E (x^2 + y^2 + z^2 ) dV where E is
bounded above by the sphere x 2 + y 2 + z 2 = 9 and below by the
cone z = √ x 2 + y 2 . i) Set up using spherical coordinates. ii)
Evaluate the integral

Use cylindrical coordinates. Where E lies below the cone ϕ = π/4
and above the sphere ρ = 1. E is a region in the first octant.

1. Evaluate ???(triple integral) E
x + y dV
where E is the solid in the first octant that lies under the
paraboloid z−1+x2+y2 =0.
2.Evaluate ???(triple integral) square root ?x^2+y^2+z^2 dV
where E lies above the cone z = square root x^2+y^2 and between
the spheres x^2+y^2+z^2=1 and x^2+y^2+z^2=9

Use cylindrical coordinates.
Evaluate
x2 + y2
dV,
E
where E is the region that lies inside the cylinder
x2 + y2 = 25 and between
the planes
z = −4
and
z = −1.

Use cylindrical coordinates.
Evaluate
6(x3 + xy2) dV, where E is
the solid in the first octant that lies beneath
the paraboloid z = 4 − x2 − y2.
E

Use cylindrical coordinates.
Evaluate the integral, where E is enclosed by the
paraboloid
z = 8 + x2 + y2,
the cylinder
x2 + y2 = 8,
and the xy-plane.
ez dV
E

Use cylindrical coordinates.
Evaluate the integral, where E is enclosed by the
paraboloid
z = 7 + x2 + y2,
the cylinder
x2 + y2 = 8,
and the xy-plane.
ez dV
E

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