Question

1)Find the centroid of the region

(Sketch the region, set up the integral. Locate the centroid in the region)(Do not answer centroid in decimals. but use decimals to locate it in the sketch.) y=(e^x)/2 y=0 x=0 x=2

2) Use the centroid to find the volume (leave the answer in terms of pi) of the region in question 1) when the region is revolved around

a)the x-axis

b) the y-axis

Answer #1

Sketch the graph. Set up (DO Not Evaluate) an integral for the
volume of the solid that results when the area bound by y=2x-x^2
and y= 0 is revolved about the y axis.

1) Set up, but do not evaluate, an integral to find the volume
when the region bounded by y=1, y=tanx and the y-axis is rotated
about the following lines:
a) The x-axis
b) The y-axis
c) The line y=2
d) The line x=3
e) The line x= -1
2) Set up, but do not evaluate, an integral to find each of the
following:
a) The volume that results when the region in the first quadrant
bounded by y=sinx, y=1 and...

Sketch the region enclosed by the equations y = tan x, y 0, x=
pi/4 . Include a typical
approximating rectangle. You may use Maple or other technology
for this.
a. Find/set-up an integral that could be used to find the area of
the region in. Do not evaluate. b. Set up an integral
that could be used to find the volume of the solid obtained by
rotating the region in #1 around the line x =pi/ 2 . Do...

Sketch and then set up the integral for the volume of revolution
of the region bounded by y= -x2+3x+4 and 3x-y=5 which is
rotated about the following line x=3

A. For the region bounded by y = 4 − x2 and the x-axis, find
the volume of solid of revolution when the area is revolved
about:
(I) the x-axis,
(ii) the y-axis,
(iii) the line y = 4,
(iv) the line 3x + 2y − 10 = 0.
Use Second Theorem of Pappus.
B. Locate the centroid of the area of the region bounded by y
= 4 − x2 and the x-axis.

#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...

1. Use the shell method to set up and evaluate the integral that
gives the volume of the solid generated by revolving the plane
region about the line x=4.
y=x^2 y=4x-x^2
2. Use the disk or shell method to set up and evaluate the
integral that gives the volume of the solid generated by revolving
the region bounded by the graphs of the equations about each given
line.
y=x^3 y=0 x=2
a) x-axis b) y-axis c) x=4

1) Find the area of the region bounded by the curves y = 2x^2 −
8x + 12 and y = −2x + 12
a) Find the volume when the area in question 1 is revolved
around the x-axis.
b)Find the volume when the area in question 1 revolved around
the y-axis.

Set up, but do not evaluate, the integral for the volume of the
solid obtained by rotating the region enclosed by y=\sqrt{x}, y=0,
x+y=2 about the x-axis. Sketch
a) By Washers
b) Cylindrical shells

R is the region bounded by ? = √? − 1, ? = 2, and the
x-axis.
a) Set up the integral you would use to find the volume of the
solid formed by revolving R around the y-axis.
b) Set up the integral you would use to find the volume of the
solid formed by revolving R around the line ? = −3
c) Set up the integral you would use to find the volume of the
solid formed...

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