Math Problem Statement

Evaluate the double integral

D

f(xy)dA

over the polar rectangular region D.

f(xy) = arctan

y

x

, where D = (r, 𝜃)  1 ≤ r ≤ 3,

𝜋

4

≤ 𝜃 ≤

𝜋

3

Solution

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Math Problem Analysis

Mathematical Concepts

Calculus
Double Integrals
Polar Coordinates
Trigonometric Functions

Formulas

f(x, y) in polar coordinates: f(x, y) = arctan(y/x) = θ
Area element in polar coordinates: dA = r dr dθ

Theorems

Conversion to Polar Coordinates
Fundamental Theorem of Calculus

Suitable Grade Level

College Level Calculus