Question

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.*

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.

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...

1. Compute for the volume by using Disk/Wasker and Shell Method
by rotating the region below around the y-axis.
y = √x, x = 2y
(a) Set up the integral using the Disk/Washer Method. Do Not
Evaluate the integral.
(b) Set up the integral using the Shell Method. Do Not Evaluate
the integral.

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...

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

Let S be the region between •x=1, •x=3, •y=6−(x−2)2, and •y=x
2+1.
a) Set up an integral to ﬁnd the area of S. Do not
evaluate.
b) Set up an integral to ﬁnd the volume Vx of the solid obtained by
rotating S about the x-axis. Do not evaluate.
c) Set up an integral to ﬁnd the volume Vy of the solid obtained by
rotating S about the y-axis. Do not evaluate.

Let R be the region bounded by y = x2 + 1, y = 0, x =
1, and x = 2. Graph the region R. Find the
volume of the solid generated when R is revolved
about the y-axis using (a) the Washer Method and
(b) the Shell Method.

1) Find the volume of the solid formed by rotating the region
enclosed by
y=e^(5x)+2, y=0, x=0, x=0.4
about the x-axis.
2) Use the Method of Midpoint Rectangles (do NOT use the
integral or antiderivative) to approximate the area under the curve
f(x)=x^2+3x+4 from x=5 to x=15. Use n=5 rectangles to find your
approximation.

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

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