Question

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

Answer #1

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

Use triple integration to find the volume of the solid cylinder
x^2 + y^2 = 9 that lies above z = 0 and below x + z = 4.

Use a triple integral to find the volume of the given solid.
The tetrahedron enclosed by the coordinate planes and the
plane
11x + y + z = 2

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...

Use triple integral and find the volume of the solid E bounded
by the paraboloid z = 2x2 + 2y2 and the plane
z = 8.

Calculate the volume with the triple integral by plotting the
region between the x² + y² + z² = 4 sphere and the plane z =
1.

Find the volume of the solid that lies under the paraboloid z =
x^2 + y^2 , above the xy-plane and inside the cylinder x^2 + y^2 =
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

Find the volume of the solid under the surface z = xy and above
the triangle with vertices (1, 1), (3, 1), and (1, 2).

A solid Tetrahedron has vertices (0, 0, 0), (2, 0, 0), (0, 4,
0), and (0, 0, 6).
(a) i. Sketch the tetrahedron in the xyz-space.
ii. Sketch (and shade) the region of integration in the
xy-plane.
(b) Setup one double integral that expresses the volume of the
tetrahedron. Define the proper limits of integration and the order
of integration. DO NOT EVALUATE.

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