Cylindrical Shells
Cylindrical Shells
Consider rotating the region between the curve
y = x^{2
} the
line
x = 2
and the xaxis about the yaxis.
If instead of taking a
cross section perpendicular to the yaxis, we take a cross section perpendicular
to the xaxis, and revolve it about the yaxis, we get a cylinder. Recall
that the area of a cylinder is given by:
where r is the radius of the cylinder and h is the height of the cylinder.
We can see that the radius is the x coordinate of the point on the
curve, and the height is the y coordinate of the curve. Hence
A(x)
= 2pxy = 2px(x^{2})
Therefore the volume is given by
Example:
Find the volume of revolution of the region bounded
by the curves
y = x^{2} + 2,
y = x + 4, and the yaxis
about the y
axis.
Solution:
We draw the picture with a cross section perpendicular to the xaxis. The
radius of the cylinder is x and the height is the difference of the
y
coordinates:
h = (x + 4)  (x^{2} +
2)
We solve for b.
(x + 4) = (x^{2} +
2)
x^{2}  x  2 = 0
(x  2)(x + 1) =
0
So that b = 2. Hence the
volume is equal to
Exercises
Find the volume of the solid formed by revolving the given region about the
given line

y = x^{2}  3x + 2, y = 0 about the yaxis

y = x^{2} 
7x + 6, y = 0 about the yaxis

x = 1  y^{2} , x = 0
(first quadrant)
about the xaxis

y = xsqrt(1 + x^{3}), y = 0, x =
2
about the yaxis

(x  1)^{2} + y^{2} = 1 about the yaxis

x^{2} + (y  1)^{2} = 1 about the xaxis

y = x^{2}  2x
+ 1, y = 1 about the line x = 3
Answers
Back
to Area and Volume Page
Back
to Math 106 Home Page
Back to the Math
Department Home
email
Questions and Suggestions
