Question

Find the integral that represents:

The volume of the solid under the cone z = sqrt(x^2 + y^2) and
over the ring 4 ≤ x^2 + y^2 ≤ 25

The volume of the solid under the plane 6x + 4y + z = 12 and
on the disk with boundary x2 + y2 = y.

The area of the smallest region, enclosed by the spiral rθ =
1, the circles r = 1 and r = 3 & the polar axis.

Answer #1

Use cylindrical coordinates.
Find the volume of the solid that is enclosed by the cone
z =
x2 + y2
and the sphere
x2 + y2 + z2 = 128.

Use polar coordinates to find the volume of the given solid.
Under the paraboloid
z = x2 + y2
and above the disk
x2 + y2 ≤ 25

Use a double integral in polar coordinates to find the volume of
the solid bounded by the graphs of the equations.
z = xy2, x2 + y2 =
25, x>0, y>0, z>0

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

a) Let R be the solid enclosed by the paraboloid z = 8 − (x^2+
y^2) and the cone z=2 sqrt(x^2+y^2) FindthevolumeofR.

find the volume between the cone z=sqrt(x^2+y^2) and the sphere
x^2+y^2+z^2=2az, if a=1

Find the mass of a thin funnel in the shape of a cone
z =
sqrt
x2 + y2
, 1 ≤ z ≤ 4
if its density function is
ρ(x, y, z) = 7 − z.

Find the integral that represents the volume of the solid
bounded by the planes y = 0, z = 0, y = x, and 6x + 2y + 3z = 6. No
need to solve the integral.

use a double integral in polar coordinates to find the volume of
the solid in the first octant enclosed by the ellipsoid
9x^2+9y^2+4z^2=36 and the planes x=sqrt3 y, x=0, z=0

Find the integral that represents the volume of the solid
bounded by the planes y = 0, z = 0, y = x and 6x + 2y + 3z = 6
using double integrals.

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