91Ó°ÊÓ

A Cartesian integral is a double integral calculated with respect to Cartesian coordinates, which are typically denoted by the variables \( x \) and \( y \). In this system, we are working on a two-dimensional plane where the x- and y-axes intersect at the origin (0,0). Cartesian integrals are often used to calculate areas under a curve or other regions defined in this coordinate system.

When setting up a Cartesian integral, it is crucial to identify the limits of integration. These limits describe the region over which the integration will be performed. For example, in the integral \( \int_{0}^{2} \int_{0}^{\sqrt{4-y^{2}}} (x^2 + y^2) \, dx \, dy \), the outer limits with respect to \( y \) span from 0 to 2, while the inner with respect to \( x \) depend on \( y \), from 0 to \( \sqrt{4-y^2} \).

This structure helps define a specific area on the plane. However, when the region is circular, converting to another system like polar coordinates can simplify the problem.

Integration is a mathematical process used to find the whole by combining pieces over specified intervals. It's essentially the reverse operation of differentiation. In the context of area calculation, integration allows us to sum up infinitesimal pieces to find the area under curves or within closed regions.

When evaluating the given integral \( \int (x^2 + y^2) \, dx \, dy \), we apply definite integration over the specified limits to find the total area or volume described by the function \( x^2 + y^2 \).

This process of integration involves finding an antiderivative of the function, substituting the limits, and calculating the resulting expression. It's important to handle integration carefully, especially with transformation techniques like switching from Cartesian to polar coordinates.

Polar transformation is a method used to convert Cartesian coordinates \((x, y)\) into polar coordinates \((r, \theta)\). This is particularly useful for dealing with problems that have circular symmetry, such as those describing regions in circles or semicircles.

In polar coordinates:

• \( x = r\cos(\theta) \)
• \( y = r\sin(\theta) \)
• Radial distance is represented by \( r \)
• Angle from the positive x-axis is \( \theta \)

By performing this transformation, the double integral \((x^2+y^2) \, dx \, dy\) converts to \( r^3 \, dr \, d\theta \). The Jacobian, which is \( r \) for this change of variables, accounts for the stretching factor that occurs during the transformation.

Therefore, our updated polar integral for this problem becomes \( \int_{0}^{\frac{\pi}{2}} \int_{0}^{2} r^3 \, dr \, d\theta \), simplifying the computation over circular regions by using forms aligned with the geometry of the problem.