Question

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

Answer #1

1- Set up the triple integral for the volume of the sphere Q=8
in rectangular coordinates.
2- Find the volume of the indicated region.
the solid cut from the first octant by the surface z= 64 - x^2
-y
3- Write an iterated triple integral in the order dz dy dx for
the volume of the region in the first octant enclosed by the
cylinder x^2+y^2=16 and the plane z=10

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

Find 6 different iterated triple integrals for the volume of the
tetrahedron cut from the first octant (when x > 0, y > 0, and
z > 0) by the plane 6x + 2y + 3z = 6. Dont evaluate the
integrals.

Use the triple integrals and spherical coordinates to find the
volume of the solid that is bounded by the graphs of the given
equations. x^2+y^2=4, y=x, y=sqrt(3)x, z=0, in first octant.

1)
Use triple integration to find the volume of a sphere with
radius 5 in cylindrical, spherical, and cartesian coordinates.
Evaluate them all.

Use a triple integral to find the volume of the solid under the
surfacez = x^2 yand above the triangle in the xy-plane with
vertices (1.2) , (2,1) and (4, 0).
a) Sketch the 2D region of integration in the xy plane
b) find the limit of integration for x, y ,z
c) solve the 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.

a
ball p = 2 is cut by the z plane = 1. If D is a smaller piece of
ball, sketch D! without calculating, determine the triple integral
and its boundaries to calculate volume D using:
a. Cartesian coordinates
b. cylinder coordinates
c. spherical coordinates

Let D be the smaller cap cut from a solid ball of radius 2 units
by a plane 1 unit from the center of the sphere. Express the volume
of D as an interated triple integral in (a) spherical and (b)
cylindrical coordinates. I was able to calculate the rectangular
coordinates, however I do not understand how I should converse
those to spherical and cylindrical coordinates.

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

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