Question

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.

Answer #1

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

Evaluate the integral ∬ ????, where ? is the square with
vertices (0,0),(1,1), (2,0), and (1,−1), by carrying out the
following steps:
a. sketch the original region of integration R in the xy-plane
and the new region S in the uv-plane using this variable change: ?
= ? + ?,? = ? − ?,
b. find the limits of integration for the new integral with
respect to u and v,
c. compute the Jacobian,
d. change variables and evaluate the...

Calculate the triple integral of xzdV where T is the solid
tetrahedron with vertices (0; 0; 0), (1; 0; 1),
(0; 1; 1), and (0; 0; 1). and please go in depth how to find
boundaries

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

find the double integral that represents the
volume of the solid entrapped by the planes y=0, z=0, y=x and
6x+2y+3z=6 (please explain how to get the limits of integration)
you don't need to solve the integral just leave it expressed

The volume of the solid obtained by rotating the region enclosed
by
?=?4?+1,?=0,?=0,?=1y=e4x+1,y=0,x=0,x=1
about the x-axis can be computed using the method of disks or
washers via an integral
?=∫??V=∫ab ? dx dy
with limits of integration ?=a= and
?=b= .
The volume is ?=V= cubic units.

The solid is a triangular column with a slanted top. The base
is the triangle in the xy-plane with sides x=0 ,
y=0 , and y=4-3x . The top is the plane
x+y-z=-2
The solid is formed by the paraboloid
f(x,y)=
x2+y2-4 and
the xy-plane. (Note that the paraboloid makes a circle as
it intersects the plane—this will become important next week
The same solid as the previous problem except now the
paraboloid is bounded above by the plane z...

4. Consider the solid bounded by the paraboloid x^2+ y^2 + z = 9
as well as by the planes y = 3x and z = 0 in the first octant.
(a) Graph the integration domain D.
(b) Calculate the volume of the solid with a double
integral.

#6) a) Set up an integral for the volume of the solid S
generated by rotating the region R bounded by x= 4y and y= x^1/3
about the line y= 2. Include a sketch of the region R. (Do
not evaluate the integral).
b) Find the volume of the solid generated when the plane region
R, bounded by y^2= x and x= 2y, is rotated about the
x-axis. Sketch the region and a typical shell.
c) Find the length of...

Consider Z 4 0 Z √ 6x−x2 √ 4x−x2 y dydx + Z 6 4 Z √ 6x−x2 0 y
dydx (a) [3 pts.] Sketch the region of integration. (b) [7 pts.]
Evaluate the integral. You may need to change the coordinate
system

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