Question

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

Answer #1

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.

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

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

Set up, but do not evaluate, the integral for the volume of the
solid obtained by rotating the region enclosed by y=\sqrt{x}, y=0,
x+y=2 about the x-axis. Sketch
a) By Washers
b) Cylindrical shells

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

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.

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.

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.

Set up an iterated integral for the triple integral in spherical
coordinates that gives the volume of the hemisphere with center at
the origin and radius 5 lying above the xy-plane.

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