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In mathematics, polar coordinates provide a way to describe a point in a plane using a distance and an angle. Rather than using traditional Cartesian coordinates (x, y), in polar coordinates, each point is determined by the radius \( r \) and the angle \( \theta \).

For the problem presented, the transformation was given by:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

This transformation is useful because it can simplify the calculation of integrals over circular regions. In polar coordinates, the area element \( d\beta^2(x,y) \) becomes \( r \, dr \, d\theta \).

By changing to polar coordinates, the integral domains and function simplified, which makes the computation more manageable. For this exercise, the region was specified by the conditions \( \frac{\pi}{4} < \theta < \frac{\pi}{2} \) and \( 0 < r < \infty \). These inequalities define exactly which part of the plane we are integrating over.

The concept of an iterated integral refers to evaluating a multiple integral by calculating one integral at a time. It breaks down a complex integral into simpler parts.

When dealing with an iterated integral, especially in two dimensions as in this problem, the idea is to compute one integral first and use its result in the computation of the second integral. In the given solution, the integral over \( r \) was evaluated first:

• \( -2 \sin(\theta) e^{-\frac{1}{2} r^2} \bigg|_{0}^{\infty} \)

Calculating this from 0 to \( \infty \), and then, the solution simplifies further.

Once the \( r \) integral was determined, it left a simpler integral over \( \theta \):

• \( 2 \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sin(\theta) d\theta \)

This structure helps ensure that integration over complex regions can be handled systematically.

Changing variables is a powerful method for simplifying complicated integrals. The technique involves substituting a variable, or set of variables, with another to make an integral easier to evaluate.

In this particular exercise, changing from Cartesian coordinates to polar coordinates transformed the problem significantly. The original two-variable function \( y e^{-\frac{1}{2}(x^2 + y^2)} \) was converted into a single-variable function by expressing \( x \) and \( y \) as \( r \cos(\theta) \) and \( r \sin(\theta) \), respectively.

This process gives you a product of functions that separate the variables, making the integration straightforward:

• The function in the new variables became \( r \sin(\theta) e^{-\frac{1}{2} r^2} \)
• The new integration elements \( dr \) and \( d\theta \) are independent, further easing the process.

This highlights how changing variables can simplify seemingly complex integration tasks, making them more tractable.