91Ó°ÊÓ

A double integral is a fundamental concept in calculus, allowing us to compute the volume under a surface in a two-dimensional region. Think of it as extending the idea of summing up areas in one-dimensional calculus to two dimensions. Double integrals are essential for calculating aspects such as mass, volume, and area in higher dimensions.
To perform a double integral, you first need to define a region in your plane where the integration will occur. In our problem, this region is defined in polar coordinates, spanning certain intervals of radial and angular values.
The double integral is often denoted as \( \iint_R f(x, y) \, dA \), where \( R \) signifies the region of integration, and \( dA \) represents a small area element. Here, the goal was to evaluate \( \iint_{R} x \, dA \), with conversion to polar coordinates.

Polar integration refers to the process of evaluating integrals where the region is described in terms of polar coordinates \((r, \theta)\) instead of Cartesian coordinates \((x, y)\). This conversion is particularly useful for regions like circles and their sectors.
To convert a Cartesian coordinate function into polar coordinates for integration, you need to use the relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( dA = r \, dr \, d\theta \)

This setup simplifies integration over circular regions or sectors.
In our task, this conversion transformed the integral \( \iint_{R} x \, dA \) into \( \iint_{R} (r \cos \theta) \, r \, dr \, d\theta \), leading to the simplified form \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_{1}^{2} r^2 \cos \theta \, dr \, d\theta \). This structured approach allowed us to evaluate the integral efficiently.

The cosine function \( \cos \theta \) represents how much the projection of a point on a circle in two dimensions lies on the x-axis. It is one of the basic trigonometric functions, characterized by its oscillating wave-like graph.
The cosine of an angle \( \theta \) is the x-coordinate of the corresponding point on the unit circle. Cosine values oscillate between -1 and 1, with specific key points such as:

• \( \cos(0) = 1 \)
• \( \cos(\frac{\pi}{2}) = 0 \)
• \( \cos(\pi) = -1 \)

These properties are crucial when evaluating integrals that involve trigonometric terms.
In polar integration, the cosine function \( \cos \theta \) was applied to simplify and solve the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{7}{3} \cos \theta \, d\theta \). Its antiderivative is \( \sin \theta \), leading to an efficient computation of \( \frac{14}{3} \) as the final result.