The Double Integral Before Iterated Integrals

1
The Double Integral Before Iterated Integrals

• Section 13.2

2
Definition of the Integral

• Multivariable Case
• If f(x,y) is defined on the closed, bounded
  region
• R, then
• provided this limit exists.

• Univariate Case
• If f(x) is defined on the closed interval a,b,
  then
• provided this limit exists.

R
3
Sufficient Conditions

• If.
• (i) f(x,y) is continuous on the closed, bounded
  region R, and
• (ii) R can be subdivided into a finite union of
  nonoverlapping regions that are vertically or
  horizontally simple,
• then f(x,y) is integrable on R.

• If f(x) is continuous on the closed interval
  a,b, then f(x) is integrable on a,b.

4
Area finding

• If f(x) is continuous on a,b and for all
  x in a,b in R, then the area A of the planar
  region that lies above the x-axis and below the
  graph of f (between a and b) is defined to be

5
Area Under a Curve
6
Volume finding

• If f(x,y) is integrable over a plane region R and
  for all (x,y) in R, then the volume V of
  the solid that lies above the region R and below
  the graph of f is defined to be

7
Volume Under a Surface f(x,y)
8
Fubini's Theorem

• Let f(x,y) be continuous on a plane region R.
  Then
• (i) If the region R is vertically simple
• (i) If the region R is horizontally simple