Evaluate a double integral over a region defined in polar coordinates by converting the integrand and area element.

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Evaluating a Double Integral over a Half-Annulus

Multiple integrals warmup

Sketch the region \( R = \{(r, \theta) \mid 1 \le r \le 2, -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\} \), and evaluate \( \iint_{R} x \, dA \).

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What you need first

Basic understanding of polar coordinates
Evaluating definite integrals
Trigonometric values at standard angles

What this problem tests

Tests the ability to convert Cartesian integrands to polar coordinates, include the correct area element, and evaluate the resulting separable double integral.

This problem is typical of an introductory multivariable calculus course when learning to set up and evaluate double integrals in polar coordinates.

Evaluating a Double Integral over a Half-Annulus

What you need first

What this problem tests

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