Question

Let R be the finite region bounded by y=x^3 and y= 9x.

(a) Write an integral that gives the volume generated by revolving R around the x-axis.

(b) Write an integral that gives the volume generated by revolving R around the line x=−9.

Answer #1

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 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

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.

Let R in the x,y-plane be in the first quadrant and bounded by
y=x+2 and y=x2, and x = 0. Find the
volume generated by revolving the region R about the line x =
4.

3.
(a) Find the volume of the solid generated by revolving the
region bounded by the graphs of the given equations around the
x-axis.
? = 0, ? = ? 3 + ?, ? = 1
(b) Find the volume of the solid generated by revolving the
region from part (a) around the y-axis

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.

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

Consider the region S enclosed by the graphs of y=x^3-6^2+9x and
y=x/2 . Determine which solid has the greater volume, and by how
much: (a) The solid generated by revolving S about the x-axis; (b)
The solid generated by revolving S about the line y=4.

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