MA 242.003
• Day 33 – February 21, 2013
• Section 12.2: Review Fubini’s Theorem
• Section 12.3: Double Integrals over General
Regions
Compute the volume below z = f(x,y)
and above the rectangle R = [a,b] x [c,d]
To be able to compute double integrals we need the concept
of iterated integrals.
Section 12.3: Double Integrals over General Regions
Section 12.3: Double Integrals over General Regions
“General Region” means a connected 2-dimensional region
in a plane bounded by a piecewise smooth curve.
Section 12.3: Double Integrals over General Regions
“General Region” means a connected 2-dimensional region
in a plane bounded by a piecewise smooth curve.
Section 12.3: Double Integrals over General Regions
Problem: Compute the double integral of f(x,y) over the
region D shown in the diagram.
Section 12.3: Double Integrals over General Regions
Problem: Compute the double integral of f(x,y) over the
region D shown in the diagram.
Solution:
Section 12.3: Double Integrals over General Regions
Problem: Compute the double integral of f(x,y) over the
region D shown in the diagram.
Solution:
Section 12.3: Double Integrals over General Regions
Problem: Compute the double integral of f(x,y) over the
region D shown in the diagram.
Section 12.3: Double Integrals over General Regions
Problem: Compute the double integral of f(x,y) over the
region D shown in the diagram.
It turns out that if we can integrate over 2 special
types of regions,
Section 12.3: Double Integrals over General Regions
Problem: Compute the double integral of f(x,y) over the
region D shown in the diagram.
It turns out that if we can integrate over 2 special
types of regions, then properties of integrals implies
we can integrate over general regions.
Some Examples:
Some Examples:
Some Examples:
Question: How do we evaluate a double integral over a type I region?
Question: How do we evaluate a double integral over a type I region?
Question: How do we evaluate a double integral over a type I region?
Question: How do we evaluate a double integral over a type I region?
(Continuation of calculation)
Example:
(continuation of example)
(continuation of example)
(continuation of example)