Question

Lets consider the solid bounded above a sphere x^2+y^2+z^2=2 and below by the paraboloid z=x^2+y^2.

Express the volume of the solid as a triple integral in cylindrical coordinates. (Please show all work clearly) Then evaluate the triple integral.

Answer #1

please comment if you have any doubts will clarify

4. Consider the solid bounded by the paraboloid x^2+ y^2 + z = 9
as well as by the planes y = 3x and z = 0 in the first octant.
(a) Graph the integration domain D.
(b) Calculate the volume of the solid with a double
integral.

write and evaluate the triple integral for the function f(x,y,z)
= z^2 bounded above by the half-sphere x^2+y^2+z^2=4 and below by
the disk x^2+y^4=4. Use spherical coordinates.

Write down a cylindrical coordinates integral that gives the
volume of the solid bounded above by z = 50 − x^2 − y^2 and below
by z = x^2 + y^2 . Evaluate the integral. (Hint: use the order of
integration dz dr dθ.)

Let S be the boundary of the solid bounded by the paraboloid
z=x^2+y^2 and the plane z=16
S is the union of two surfaces. Let S1 be a portion of the plane
and S2 be a portion of the paraboloid so that S=S1∪S2
Evaluate the surface integral over S1
∬S1 z(x^2+y^2) dS=
Evaluate the surface integral over S2
∬S2 z(x^2+y^2) dS=
Therefore the surface integral over S is
∬S z(x^2+y^2) dS=

The solid bounded below the sphere ? = 1 and above by the
Cardioid revolution ? = 1 + cos?. a) Find the volume of the solid.
b) Set up the cylindrical integral for finding the average value of
function ?(?,?, ?) = 2? over the solid. Do Not evaluate it.

The domain E of R^3 located inside the sphere x^2 + y^2 + z^2 =
12 and above half-cone z = sqrroot(( x^2 + y^2) / 3) (a) Represent
the domain E.
(b) Calculate the volume of solid E with a triple integral in
Cartesian coordinates.
(c) Recalculate the volume of solid E using the cylindrical
coordinates.

Set up a triple integral in cylindrical coordinates to compute
the volume of the solid bounded between the cone z 2 = x 2 + y 2
and the two planes z = 1 and z = 2.
Note: Please write clearly. That had been a big problem for me
lately. no cursive Thanks.

Set up (Do Not Evaluate) a triple integral that yields the
volume of the solid that is below
the sphere x^2+y^2+z^2=8
and above the cone z^2=1/3(x^2+y^2)
a) Rectangular coordinates
b) Cylindrical
coordinates
c) Spherical
coordinates

Use a triple integral in cylindrical coordinates to find the
volume of the sphere x^2+ y^2+z^2=a^2

Find the volume of the region bounded below by the paraboloid z
= x^2 + y^2 and above by the plane z = 2x.

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