Question

**Let R be the region bounded by y=ln(x), the x-axis, and
the line x=e. Find the volume of the solid that is generated when
the region R is revolved about the x-axis.**

Answer #1

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.

Let R be the region bounded by the curves y = x, y = x+ 2, x =
0, and x = 4. Find the volume of the solid generated when R is
revolved about the x-axis. In addition, include a carefully labeled
sketch as well as a typical approximating disk/washer.

A. For the region bounded by y = 4 − x2 and the x-axis, find
the volume of solid of revolution when the area is revolved
about:
(I) the x-axis,
(ii) the y-axis,
(iii) the line y = 4,
(iv) the line 3x + 2y − 10 = 0.
Use Second Theorem of Pappus.
B. Locate the centroid of the area of the region bounded by y
= 4 − x2 and the x-axis.

Let X be the region bounded by y=x and y=x^2.
Find the volume of the solid that is made when X is revolved
about the y-axis. Find it in two different methods, include a
diagram for each method, and explain.

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 bounded by 6 cos x, y = e^x , x = 0, and ?
=pi /2 . Using a method of your choice, find the volume of the
solid generated by revolving R around the line y = 7. Give an exact
numerical answer.

The region R is bounded by y=x^2 and y=sqrtx. Find the volume of
the solid generated by revolving R about
a) the x-axis b) the y-axis c) the line y=1 d) the line y=-1 e)
the line x=1 f) the line x=-1

the region bounded by the curves xy=4 and x+y=15 is revolved
about y axis. calculate the volume of the solid generated
full calculation

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 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

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