Question

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.

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, but do not evaluate, an iterated integral in polar
coordinates for ∬ 2x + y dA where R is the region in the xy-plane
bounded by y = −x, y = (1 /√ 3) x and x^2 + y^2 = 3x. Include a
labeled, shaded, sketch of R in your work.

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

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

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.

Set up and evaluate a triple integral for the volume of a
cylinder of radius a and height h. You need to write an equation
for this cylinder first

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

Set up an iterated integral for the surface area of the part of
the plane x + y + z = 6 that lies in the first octant.

use spherical coordinates to calculate the triple integral of
?(?, ?, ?)=?2+?2 over the region ?≤8
∫∫∫?(?^2+?^2) ??

Write a triple integral including limits of integration that
gives the volume of the cap of the solid sphere x2+y2+z2≤34 cut off
by the plane z=5 and restricted to the first octant. (In your
integral, use theta, rho, and phi for θ, ρ and ϕ, as needed.)

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