Question

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.

Answer #1

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

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.)

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

1a. Using rectangular coordinates, set up iterated integral that
shows the volume of the solid bounded by surfaces z= x^2+y^2+3,
z=0, and x^2+y^2=1
1b. Evaluate iterated integral in 1a by converting to polar
coordinates
1c. Use Lagrange multipliers to minimize f(x,y) = 3x+ y+ 10 with
constraint (x^2)y = 6

a) Set up the integral in cylindrical coordinates for the moment
of inertia about the z-axis for a cone bounded by z = sqrt x^2+y^2
and z = 2. The density of the cone is a constant 5.
b) Evaluate the integral by using direct method the work done by ~
F(x,y) = (y,−x) along ~r(t) = (2cos(t),2sin(t)), 0 ≤ t ≤ π.
need help on a and b

Use cylindrical coordinates to find the volume of the solid
bounded by the graphs of z = 68 − x^2 − y^2 and z = 4.

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.

Set up a double integral in rectangular coordinates for the
volume bounded by the cylinders x^2+y^2=1 and y^2+z^x=1 and
evaluate that double integral to find the volume.

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.

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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