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.)

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.

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

valuate SSSEz^2dV, where E is the solid region bounded below by
the cone z=2sqr(x^2+y^2) and above by plane z=10.
(SSS) = Triple Integral

Find the volume of the solid which is bounded by the cylinder
x^2 + y^2 = 4 and the planes z = 0 and z = 3 − y. Partial credit
for the correct integral setup in cylindrical coordinates.

Use triple integral and find the volume of the solid E bounded
by the paraboloid z = 2x2 + 2y2 and the plane
z = 8.

Find the volume of the solid using triple integrals. The solid
region Q cut from the sphere x^2+y^2+z^2=4 by the cylinder r=2sinϑ.
Find and sketch the solid and the region of integration R. Setup
the triple integral in Cartesian coordinates. Setup the triple
integral in Spherical coordinates. Setup the triple integral in
Cylindrical coordinates. Evaluate the iterated integral

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

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