Question

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

Answer #1

we can plot the given region as follows,

Consider a shell at a distance x from the y-axis.

Then the distance b/w the shell and axis of rotation is the radius of rotation given by,

The height of the shell is given by,

The volume of rotation about the vertical line using the shell method is given by,

plugin the values,

plugin the limits,

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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 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 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...

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 answer all question explain. thank you.
(1)Consider the region bounded by y= 5- x^2 and y = 1. (a)
Compute the volume of the solid obtained by rotating this region
about the x-axis.
(b) Set up the integral for the volume of the solid obtained by
rotating this region about the line x = −3. No need to evaluate the
integral, just set it up.
(2) (a) Find the exact (no calculator approximation) average
value of the function f(x)...

Consider the region bounded by y=sqrt(x) and y=x^3
a) Find the area of this region
b) Find the volume of the solid generated by rotating this
region about the x-axis using washer
c) Find the volume of the solid generated by rotating this
region about the horizontal line y=3 using shells

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.

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...

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.

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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