Substitutions in Multiple Integrals

Multiple integration is an important mathematical method used to calculate physical quantities distributed over a two-dimensional region, such as the total mass of an elliptical plate. In this process, the density function is evaluated throughout the entire region enclosed by the ellipse. The contributions from all points inside the boundary are then accumulated to determine the total mass.

When integration is performed directly in rectangular coordinates, the elliptical boundary produces limits involving complicated square root expressions. These variable limits make the integration process difficult and less efficient. As a result, direct evaluation of the integral becomes cumbersome, especially for nonuniform density functions.

To simplify the problem, a change of variables is introduced. By scaling the coordinate axes appropriately, the elliptical region is transformed into a unit circle in a new coordinate system. This transformation converts the complicated elliptical boundary into a simpler circular boundary with constant limits of integration. The geometric transformation also changes the measurement of area within the region.

Because the transformation stretches or compresses the region, a Jacobian determinant is required as a correction factor. The Jacobian accounts for the change in area caused by the coordinate transformation and ensures that the transformed integral accurately represents the original physical region.

After transforming the ellipse into a circular region, polar coordinates are introduced to simplify the integration further. Polar coordinates are naturally suited for circular domains because the boundary can be expressed using constant radial and angular limits. This transformation introduces a second Jacobian factor associated with the geometry of circular sectors.

By combining the coordinate transformations with the appropriate Jacobian correction factors, the original integral over the elliptical region is converted into a much simpler integral over a circular region. The resulting integral has constant limits and is easier to evaluate, demonstrating the usefulness of coordinate transformations in multiple integration problems.