Find the volume under a rational surface over a circular sector in the first quadrant using polar coordinates.

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Volume Under a Surface Using Polar Coordinates

Multiple integrals core

Find the volume under the surface $f(x, y) = \frac{1}{x^2 + y^2 + 1}$ over the sector of the circle with radius $a$ centered at the origin in the first quadrant, as shown in Figure 13.3.4.

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What you need first

Evaluating double integrals
Converting Cartesian coordinates to polar coordinates
Integration by substitution
The area element in polar coordinates

What this problem tests

Tests the ability to set up a double integral in polar coordinates, including finding the correct bounds and using the polar area element.

This problem is typical of a multivariable calculus course when introducing double integrals in polar coordinates.

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