Double Integrals in Polar Coordinates: Evaluating Integrals and Finding Areas and Volumes, Study notes of Mathematics

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Math 114-004, Fall 2009

Tong Zhu

Department of Mathematics University of Pennsylvania

November 12, 2009

Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates

Example 1 Evaluate

D xydA^ where^ D^ is the disk centered at the origin with radius 3.

Polar Coordinate System

Polar Coordinate System

x = r cos θ y = r sin θ

How can we use polar coordinates to evaluate ∫ ∫

D

f (x, y )dA?

How can we use polar coordinates to evaluate ∫ ∫

D

f (x, y )dA?

This is in fact a special case of change of variables in multiple integrals, which will be shown in later section.

Geometrically, what is dA in the polar coordinates?

How can we use polar coordinates to evaluate ∫ ∫

D

f (x, y )dA?

This is in fact a special case of change of variables in multiple integrals, which will be shown in later section.

Geometrically, what is dA in the polar coordinates?

How can we use polar coordinates to evaluate ∫ ∫

D

f (x, y )dA?

This is in fact a special case of change of variables in multiple integrals, which will be shown in later section.

Geometrically, what is dA in the polar coordinates?

dA = rdrdθ ⇒

D

f (x, y )dA =

D

f (r cos θ, r sin θ)r drdθ

Revisit Example 1 Evaluate

D xydA^ where^ D^ is the disk centered at the origin with radius 3.

Example 3 Find the volume of the solid under the cone z =

x^2 + y 2 and above the disk x^2 + y 2 ≤ 4.