Question

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 which computes the volume of the solid obtained by rotating this region about

the y-axis using the cross-section method.

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...

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)...

Consider the plane region R bounded by the curve y = x − x 2 and
the x-axis. Set up, but do not evaluate, an integral to find the
volume of the solid generated by rotating R about the line x =
−1

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

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.

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.

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

Consider the region in the xy-plane that is bounded by y =
2x^2and y = 4x for x ≥ 0.Revolve this region about the y-axis
superfast so that your eyes glaze over and it looks like a solid
object. Find the volume of this solid.

Consider the solid obtained by rotating the region bounded by
the given curves about the x-axis. y = 5 x^3 y = 5 x x >= 0 Find
the volume V of this solid. V =

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