Question

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.

Answer #1

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.

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.)

Please answer all question explain. thank you.
(1)Consider the region bounded by y= 5- x^2 and y = 1. (a)
Compute the volume of the solid obtained by rotating this region
about the x-axis.
(b) Set up the integral for the volume of the solid obtained by
rotating this region about the line x = −3. No need to evaluate the
integral, just set it up.
(2) (a) Find the exact (no calculator approximation) average
value of the function f(x)...

a) Sketch the solid in the first octant bounded by:
z = x^2 + y^2 and x^2 + y^2 = 1,
b) Given
the volume density which is proportional to the distance from the
xz-plane, set up integrals
for finding the
mass of the solid using cylindrical
coordinates, and spherical coordinates.
c) Evaluate one of these to find the mass.

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

The region is bounded by y=2−x^2 and y=x. (a) Sketch the region.
(b) Find the area of the region. (c) Use the method of cylindrical
shells to set up, but do not evaluate, an integral for the volume
of the solid obtained by rotating the region about the line x = −3.
(d) Use the disk or washer method to set up, but do not evaluate,
an integral for the volume of the solid obtained by rotating the
region about...

Let E be the solid that lies in the first octant, inside the
sphere x2 + y2 + z2 = 10. Express the volume of E as a triple
integral in cylindrical coordinates (r, θ, z), and also as a triple
integral in spherical coordinates (ρ, θ, φ). You do not need to
evaluate either integral; just set them up.

The region is bounded by y = 2 − x^ 2 and y = x
Use the method of cylindrical shells to set up, but do not
evaluate, an integral for the volume of the solid obtained by
rotating the region about the line x = −3

draw the solid bounded above z=9/2-x2-y2
and bounded below x+y+z=1. Find the volume of this
solid.

3. Find the volume of the solid of revolution. The region is
bounded by y= 4x and y = x^3 and x ≥ 0.
a) Make a sketch.
b) About the x axis (disk/washer method).
c) About the x axis (cylindrical shells).
d) About the y axis (disk/washer method).
e) About the y axis (cylindrical shells).

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