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Polar coordinates are a method of representing points in a plane using a distance and an angle. In contrast to the rectangular coordinate system (which uses an x-coordinate and a y-coordinate), polar coordinates specify locations by a radius (\( r \)) and an angle (\( \theta \)).

• The radius (\( r \)) is the distance from the origin to the point.
• The angle (\( \theta \)) is measured in radians or degrees, starting from the positive x-axis, counter-clockwise.

Polar coordinates are especially useful in situations involving circular or spiral patterns, where using rectangular coordinates would be cumbersome. Converting between polar and rectangular coordinates involves the relationships:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• The reverse transformation is given by:
  • \( r = \sqrt{x^2 + y^2} \)
  • \( \theta = \text{tan}^{-1}(\frac{y}{x}) \)

Understanding polar coordinates simplifies solving problems involving radial symmetry, like the one above, where the computation within a specific circular region is needed.

Double integrals are a core tool in multivariable calculus, allowing us to integrate functions of two variables over a given region.This concept extends the idea of a single integral to two dimensions, making it useful for calculating areas, volumes, and other quantities that accumulate over a 2D space.Consider a region \( D \) in the xy-plane and a function \( f(x, y) \). The double integral of \( f \) over \( D \) is written as:\[\int \int_{D} f(x, y) \, dA\]where \( dA \) represents an infinitesimal element of area in \( D \).In the context of polar coordinates, the double integral takes a slightly altered form:\[\int_{r_1}^{r_2} \int_{\theta_1}^{\theta_2} f(r, \theta) r \, d\theta \, dr\]Here,

• The presence of \( r \, d\theta \, dr \) accounts for the change in area element size when transforming from Cartesian to polar coordinates.
• \( r \theta \) themselves define the bounds of integration instead of traditional x and y limits.

Double integrals are therefore essential for evaluating areas and other properties over specified regions, as demonstrated in this problem.

The region of integration defines the specific area within which an integral is evaluated. To solve a double integral problem, one must accurately sketch and understand this region in the given coordinate system, in this case, polar coordinates.

• In polar coordinates, thee region is defined by both radius limits and angle limits.
• The given limits for radius are \( r = 3 \) to \( r = 4 \), creating two concentric circles.
• The angle varies from \( \theta = \frac{3\pi}{4} \) to \( \theta = \frac{3r}{2} \).

To visualize the region of integration:

• Draw two circles centered at the origin with radii 3 and 4.
• Draw a line at \( \theta = \frac{3\pi}{4} \), marking the beginning of the sector.
• For each \( r \), sketch lines at \( \theta = \frac{3r}{2} \) until the outer circle is reached.

This approach results in a segment that begins at \( 135 \) degrees and expands as \( r \) increases, reaching the boundaries determined by \( \theta = \frac{3r}{2} \).Understanding this region is crucial for correctly evaluating the integral and accurately applying the double integral boundaries.