Question

1. The graph of y = f(x) = 1/ x^2 , 1 ≤ x < ∞, is rotated around the x-axis.

(a) Calculate the resulting volume.

(b) Set up the integral that gives the resulting surface area, but DO NOT WORK OUT THE INTEGRAL.

Answer #1

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.

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.

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...

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...

1. Suppose you have a circular tank with a 3 meter diameter and
2 meter height that is completely filled with water. Set up, DO NOT
EVALUATE, a single integral to find the work required to empty the
entire tank. Note that water weighs 9800 kg/m3 .
2.Consider the region bounded by the curves y =x^2 and y = 3x is
rotated about the x-axis. Use the disc/washer method to set up, DO
NOT EVALUATE, a single integral to find...

If 0≤f(x)≤g(x) for x in the interval [a, b],can the surface
obtained by rotating the graph of y=g(x) around the x-axis over the
interval have less surface area than the surface obtained by
rotating the graph of y=f(x) around the x-axis over the same
interval?
I have heard yes and also no so I would like some clarification.
Thanks

y=6-2x-x^2 and y=x+6, about the x-axis. Draw the figure,
set up the integral needed to find the volume when the given curve
is rotated about the specified axis, find the answer. Use
calculator if needed!

Sketch the graph. Set up (DO Not Evaluate) an integral for the
volume of the solid that results when the area bound by y=2x-x^2
and y= 0 is revolved about the y axis.

1)Find the volume of the solid whose base is a circle with
equation x^2+y^2=36 and cross-sections are squares perpendicular to
the x-axis.
(a) Create the graph for this problem
(b) What is the volume of one 'slice'?
(c) What is the integral for the volume?
(d) What is the volume in exact form?
2) Find the volume of the region bounded by y=-x^2+4 and y=x+2
rotated about the line y=5
(a) Create the graph for this problem
(b) What is...

1. Find the area between the curve f(x)=sin^3(x)cos^2(x) and y=0
from 0 ≤ x ≤ π
2. Find the surface area of the function f(x)=x^3/6 + 1/2x from
1≤ x ≤ 2 when rotated about the x-axis.

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