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 conceptof 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?

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Example:

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