Question

2. Set up an integral to find the volume of the solid generated
when the region bounded by y = 2x^{2} and y = x^{3}
is

(a) Rotated about the x-axis using shells

(b) Rotated about the x-axis using washers

(c) Rotated about the y-axis using shells

(d) Rotated about the y-axis using washers

(e) Rotated about the line x = −3

(f) Rotated about the line y = −2

(g) Rotated about the line y = 11

(h) Rotated about the line x = 7

Answer #1

Set up an integral to ﬁnd the volume of the solid generated when
the region bounded by y = square root(x) and y = (1/2)x is
(a) Rotated about the x-axis using washers
(b) Rotated about the line x = −4 using washers
(c) Rotated about the line y = 5 using shells

1) Set up, but do not evaluate, an integral to find the volume
when the region bounded by y=1, y=tanx and the y-axis is rotated
about the following lines:
a) The x-axis
b) The y-axis
c) The line y=2
d) The line x=3
e) The line x= -1
2) Set up, but do not evaluate, an integral to find each of the
following:
a) The volume that results when the region in the first quadrant
bounded by y=sinx, y=1 and...

#6) a) Set up an integral for the volume of the solid S
generated by rotating the region R bounded by x= 4y and y= x^1/3
about the line y= 2. Include a sketch of the region R. (Do
not evaluate the integral).
b) Find the volume of the solid generated when the plane region
R, bounded by y^2= x and x= 2y, is rotated about the
x-axis. Sketch the region and a typical shell.
c) Find the length of...

Set up the integral (do not evaluate) to find the volume of the
solid generated by revolving the region about the line x=5.
The region is bounded the graphs x=y^2, x=4
Use the disk and shell methods.

Set up, but do not evaluate, the integral for the volume of the
solid obtained by rotating the region enclosed by y=\sqrt{x}, y=0,
x+y=2 about the x-axis. Sketch
a) By Washers
b) Cylindrical shells

Set up an integral using the shell method to find the volume of
the solid generated by revolving the area bounded by the graphs of
the equations y=1/(x+2) y =0, x = 0 and x=5 about the line x=
-3.Sketch a graph of the region, highlighting your slice

Set up, but do not evaluate, an integral to find each of the
following:
a) The volume that results when the region in the first quadrant
bounded by y=sinx, y=1 and the y-axis is rotated about the
x-axis.
b) The volume that results when the region that is bounded by
y=x3 , y=8 and the y-axis is rotated about the y-axis. c) The
volume when the region bounded by y=ex , x=1, the x-axis and the
y-axis is rotated about...

1. The region bounded by y=x8 and y=sin(πx/2) is
rotated about the line x=−7.
Using cylindrical shells, set up an integral for the volume of the
resulting solid.
2.The region bounded by y=9/(1+x2), y=0, x=0 and x=8
is rotated about the line x=8.
Using cylindrical shells, set up an integral for the volume of the
resulting solid.

Find the volume of the solid generated by revolving the following
region about the given axis. The region in the first quadrant
bounded above by the curve y=x^2, below the x-axis, and on the
right by the line x=1, about the line x=-4. Use the washer method
to set up the integral that gives the volume of the solid.

1. Use the shell method to set up and evaluate the integral that
gives the volume of the solid generated by revolving the plane
region about the line x=4.
y=x^2 y=4x-x^2
2. Use the disk or shell method to set up and evaluate the
integral that gives the volume of the solid generated by revolving
the region bounded by the graphs of the equations about each given
line.
y=x^3 y=0 x=2
a) x-axis b) y-axis c) x=4

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