5.1
5.1
Volumes
1
Contemporary Calculus
VOLUMES OF SOLIDS
The last chapter emphasized a geometric interpretation of definite integrals as "areas" in two dimensions.
This section emphasizes another geometrical use of integration, calculating volumes of sol

8.1 Improper Integrals
Contemporary Calculus
1
8.1 Improper Integrals
b
Our original development of the definite integral
f(x) dx used Riemann sums and assumed that
a
the length of the interval of integration [a, b] was finite and
that f(x) was defined a

Volumes of Solids of Revolution
Calculus II
Some Gifs
http:/www.mathdemos.org/mathdemos/washerm
ethod/gallery/gallery.html
Disks and Washers
Revolving
around the x axis:
Disks: (touches x axis)
Washers: (doesnt touch x axis)
Example 1
Find
the volume of

CALCULUS II FORMULA SHEET FOR TEST TWO
Revolving around the x axis:
(touches x axis)
(f ( x )2 dx
(doesnt touch x axis)
( top 2bottom2 ) dx
Revolving around the y axis:
(touches x axis)
2 x f ( x ) dx
(doesnt touch x axis)
2 x ( TopBottom ) dx
Arc Lengt