Calculus 3
DISCUSS
WORK ON
PRESENT
ASSIGNMENT
Tuesday, November 25
Volumes over more general regions
97, 101, 102
97, 101
Write up 102
Skip 3.4
Read 4.1
Calendar:
Monday, December 1:
Tuesday, December 2:
Thursday, December 4:
Friday, December 5:
Wednesday, December 10:
4.1
4.3
Review
Review
Oral Final
Volumes over more general regions
We can find volumes over rectangles a < x < b, c < y < d
What about volumes over non-rectangular regions?
Type I regions
z = f(x,y)
a < x < b,
g(x) < y < h(x)
Type II regions
z = f(x,y)
g(y) < x < h(y),
c<y<d
dy dx
dx dy
Many problems are only type I or else only type II, but
some problems can be considered either way. Sometimes
the math is easier one way.
Example: Consider the volume bounded by z = 4 – x – 2y
and the coordinate planes.
a) Set up integrals for a type I solution
b) Set up integrals for a type II solution