Question

Consider the region bounded by ? = 4? , ? = 1 and x-axis. Set
up the appropriate integrals

for finding the volumes of revolution using the specified
method and rotating about the specified axis. Be sure to first
sketch the region and draw a typical cross section. SET UP THE
INTEGRALS ONLY. DO NOT evaluate the integral.

a) Disc/washer method about the x-axis

b) Shell method about the y-axis

c) Disc/washer method about the line ? = 2.

d) Shell method about the line ? = -1

Answer #1

Consider the region in the xy-plane bounded by the curves y =
3√x, x = 4 and y = 0.
(a) Draw this region in the plane.
(b) Set up the integral which computes the volume of the solid
obtained by rotating this region about
the x-axis using the cross-section method.
(c) Set up the integral which computes the volume of the solid
obtained by rotating this region about
the y-axis using the shell method.
(d) Set up the integral...

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...

The region is bounded by y=2−x^2 and y=x. (a) Sketch the region.
(b) Find the area of the region. (c) Use the method of cylindrical
shells to set up, but do not evaluate, an integral for the volume
of the solid obtained by rotating the region about the line x = −3.
(d) Use the disk or washer method to set up, but do not evaluate,
an integral for the volume of the solid obtained by rotating the
region about...

Let R be the region bounded by y = ln(x), the x-axis, and the
line x = π.
a.Usethecylindrical shell method to write a deﬁnite integral
(BUTDONOTEVALUATEIT) that gives the volume of the solid obtained by
rotating R around y-axis
b. Use the disk (washer) method to write a deﬁnite integral (BUT
DO NOT EVALUATE IT) that gives the volume of the solid obtained by
rotating R around x-axis.

Sketch the region R bounded by the graph of y=1-(x+2)^2 and the
x-axis. Set up, but do not evaluate, definite integrals that can be
used to find the volumes of the solids obtained by revolving R
about the given lines. In each case state whether your solution
primarily uses the method of disks, washers, or shells.
a) y=0
b)x=0
c)x=4
d)y=-5

Set up the integral. you do NOT have to
integrate.
A region is bounded by the x-axis and y = sinx with 0 ≤
x ≤ π. A solid of revolution is obtained by rotating that region
about the line y =−2.

Find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line, using the
disk/washer method. Sketch the region, the solid, and a typical
disk or washer.

Find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified axis. Your set-up
must include a graph of the region and an example of the rectangle
used or one disk/washer or shell (depending on the method
used).
y = SQRT(x-1) x=5, y=0, about the y-axis

Consider the region R bounded by y = sinx, y = −sinx , from x =
0, to x=π/2.
(1) Set up the integral for the volume of the solid obtained by
revolving the region R around
x = −π/2
(a) Using the disk/washer method.
(b) Using the shell method.
(2) Find the volume by evaluating one of these integrals.

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

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