Page 238 - Calculus Workbook For Dummies

222 Part IV: Integration and Infinite Series

Volumes of Weird Solids: No, You’re

Never Going to Need This

Integration works by cutting something up into an infinite number of infinitesimal
pieces and then adding the pieces up to compute the total. In this way, integration is
able to determine the volume of bizarre shapes: It cuts the shapes up into thin pieces
that have ordinary shapes which can then be handled by ordinary geometry. This sec-
tion shows you three different methods:

The meat slicer method: This works just like a deli meat slicer — you cut a
shape into flat, thin slices. You then add up the volume of the slices. This
method is used for odd, sometimes asymmetrical shapes.
The disk/washer method: With this method, you cut up the given shape into
thin, flat disks or washers (you know — like pancakes or squashed donuts). Used
for shapes with circular cross-sections.
The cylindrical shell method: Here, you cut your volume up into thin nested
shells. Each one fits snugly inside the next widest one, like telescoping tubes or
nested Russian dolls. Also used for shapes with circular cross-sections.

Q. What’s the volume of the shape shown in A. The volume is ⁄5 cubic units.
2
the following figure? Its base is formed by 1. Always try to sketch the figure first (of
the functions f x = x and g x = - x . course, I’ve done it for you here).
^ h
^ h
Its cross-sections are isosceles triangles
whose heights grow linearly from zero at 2. Indicate on your sketch a representative
the origin to 1 when x = 1. thin slice of the volume in question.
This slice should always be perpendicu-
f(x) = √x
lar to the axis or direction along which
h=1 you are integrating. In other words, if
your integrand contains, say, a dx, your
y
slice should be perpendicular to the
x-axis. Also, the slice should not be at
(1, 1)
either end of the 3-dimensional figure
or at any other special place. Rather, it
should be “in the middle of nowhere.”
x
(1, 0)
3. Express the volume of this slice.
It’s easy to show — trust me — that the
(1, –1)
height of each triangle is the same as its
g(x) = –√x x-coordinate. Its base goes from - x up
to x and is thus 2 x. And its thickness
is dx.

Therefore,
1
$
=
Volume slice = ` 2 x x dx x x dx
j
2
4. Add up the slices from 0 to 1 by
integrating.
1 1
# x x dx = 2 x E = 2 cubic units
/ 5 2
5 5
0 0