Question

Problem (9). Let R be the region enclosed by y = 2x, the x-axis,
and x = 2. Draw the solid and set-up an integral (or a sum of
integrals) that computes the volume of the solid obtained by
rotating R about:

(a) the x-axis using disks/washers

(b) the x-axis using cylindrical shells

(c) the y-axis using disks/washer

(d) the y-axis using cylindrical shells

(e) the line x = 3 using disks/washers

(f) the line y = 4 using cylindrical shells

Answer #1

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...

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.

Let R be the region in enclosed by y=1/x, y=2, and x=3. a)
Compute the volume of the solid by rotating R about the x-axis. Use
disk/washer method. b) Give the definite integral to compute the
area of the solid by rotating R about the y-axis. Use shell
method. Do not evaluate the integral.

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...

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...

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 region is bounded by y = 2 − x^ 2 and y = x
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

PLEASE USE TWO dy-INTEGRALS
Let R be the region in the first quadrant bounded by the curves
y = f(x) = 2x+ 1 and y = g(x) = 2x 2 − 8x + 9. Find the
volume of the solid obtained by rotating the region R about y-axis
using two dy-integrals.

Let B be the region bounded by the part of the curve y = sin x,
0 ≤ x ≤ π, and the x-axis. Express (do not evaluate) the volume of
the solid obtained by rotating the region B about the y-axis as
definite integrals
a) using the cylindrical shell method
b) using the disk method

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.

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