Question

Use spherical coordinates. Evaluate (6 − x^2 − y^2) dV, where H is the solid hemisphere x^2 + y^2 + z^2 ≤ 16, z ≥ 0.

Answer #1

Use spherical coordinates.
(a) Find the centroid of a solid homogeneous hemisphere of
radius 1. (Assume the upper hemisphere of a sphere centered at the
origin. Use the density function
ρ(x, y,
z) = K.
(x, y, z) =
(b) Find the moment of inertia of the solid in part (a) about a
diameter of its base.
Id =

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

Evaluate the triple integrals E y2 dV, where E is the solid
hemisphere x2 + y2 + z2 ≤ 9, y ≤ 0.
Calculus 3 Multivarible book James Stewart Calculus Early
Transcendentals 8th edition 15.8

) Use spherical coordinates to find the volume of the solid
situated below x^2 + y ^2 + z ^2 = 1 and above z = sqrt (x ^2 + y
^2) and lying in the first octant.

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.

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

Evaluate the triple integral _ D sqrt(x^2+y^2+z^2) dV, where D
is the solid region given by 1 (less than or equal to) x^2+y^2+z^2
(less than or equal to) 4.

4. Let W be the three dimensional solid inside the sphere x^2 +
y^2 + z^2 = 1 and bounded by the planes x = y, z = 0 and x = 0 in
the first octant. Express ∫∫∫ W z dV in spherical coordinates.

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.

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