Question

a. Let S be the solid region first octant bounded by the coordinate planes and the planes x=3, y=3, and z=4 (including points on the surface of the region). Sketch, or describe the shape of the solid region E consisting of all points that are at most 1 unit of distance from some point in S. Also, find the volume of E.

b. Write an equation that describes the set of all points that are equidistant from the origin and the point (2,−1,−2). What does this set look like?

Answer #1

The tetrahedron is the first octant bounded by the coordinate
planes and the plane passing through (1,0,0), (0,2,0), and
(0,0,3).
I need to calculate the volume of this region, how should this
be done?

6. Let R be the tetrahedron in the first octant bounded by the
coordinate planes and the plane passing through (1, 0, 0), (0, 1,
0), and (0, 0, 2) with equation 2x + 2y + z = 2, as shown below.
Using rectangular coordinates, set up the triple integral to find
the volume of R in each of the two following variable orders, but
DO NOT EVALUATE.
(a) triple integral 1 dxdydz
(b) triple integral of 1 dzdydx

B is the solid occupying the region of the space in the first
octant and bounded by the paraboloid z = x2 + y2- 1 and the planes
z = 0, z = 1, x = 0 and y = 0. The density of B is proportional to
the distance at the plane of (x, y).
Determine the coordinates of the mass centre of solid B.

Let D be the solid in the first octant bounded by the planes
z=0,y=0, and y=x and the cylinder 4x2+z2=4.
Write the triple integral in all 6 ways.

a) Sketch the solid in the first octant bounded by:
z = x^2 + y^2 and x^2 + y^2 = 1,
b) Given
the volume density which is proportional to the distance from the
xz-plane, set up integrals
for finding the
mass of the solid using cylindrical
coordinates, and spherical coordinates.
c) Evaluate one of these to find the mass.

Find the volume of the solid bounded by the surface z=
5+(x-y)^2+2y and the planes x = 3, y = 3 and coordinate planes.
a. First, find the volume by actual calculation.
b. Estimate the volume by dividing the region into nine equal
squares and evaluating the functional value at the mid-point of the
respective squares and multiplying with the area and summing it.
Find the error from step a.
c. Then estimate the volume by dividing each sub-square above...

a.) Let S be the solid obtained by rotating the region bounded
by the curves y=x(x−1)^2 and y=0 about the y-axis. If you sketch
the given region, you'll see that it can be awkward to find the
volume V of S by slicing (the disk/washer method). Use cylindrical
shells to find V
b.) Consider the curve defined by the equation xy=12. Set up an
integral to find the length of curve from x=a to x=b. Enter the
integrand below

(a) Find the volume of the solid generated by revolving the
region bounded by the curves y=x2 andy=4x−x2 abouttheliney=6.
[20marks]
(b) Sketch the graph of a continuous function f(x) satisfying
the following properties: (i) the graph of f goes through the
origin
(ii) f′(−2) = 0 and f′(3) = 0.
(iii) f′(x) > 0 on the intervals (−∞,−2) and (−2,3).
(iv) f′(x) < 0 on the interval (3,∞).
Label all important points. [30 marks]

Let R be the region of the plane bounded by y=lnx and the x-axis
from x=1 to x= e. Draw picture for each
a) Set up, but do not evaluate or simplify, the definite
integral(s) that computes the volume of the solid obtained by
rotating the region R about they-axis using the disk/washer
method.
b) Set up, but do not evaluate or simplify, the definite
integral(s) that computes the volume of the solid obtained by
rotating the region R about...

#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...

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