91Ó°ÊÓ

To understand triple integrals in spherical coordinates, imagine slicing through a sphere in three dimensions, much like peeling an orange. In spherical coordinates, we represent points using

• \( \rho \), the radius: distance from the origin
• \( \phi \), the polar angle: angle from the positive z-axis
• \( \theta \), the azimuthal angle: rotation around the z-axis

Switching from Cartesian to spherical coordinates often simplifies integration, especially in cases involving symmetrical shapes such as spheres. The mathematical relationships are:

• \( x = \rho \sin \phi \cos \theta \)
• \( y = \rho \sin \phi \sin \theta \)
• \( z = \rho \cos \phi \)

When a function or region of integration is symmetrical around an axis, spherical coordinates can make the integration process more straightforward, reducing complex geometric regions into simpler, radial components. In spherical coordinates, the volume element \( dV \) is represented by \( \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \), owing to the Jacobian transformation.

In solving a triple integral, one must first determine the bounds and limits of integration that define the region, often referred to as the region of integration. This region, denoted as \( B \) in our problem, is given as the upper half of a sphere. More specifically, it is bound by \( x^2 + y^2 + z^2 \leq 90 \) and \( z \geq 0 \). These inequalities define a 3D solid that is symmetric around the center and only extends above the xy-plane.
This region is crucial because it determines the values that the variables within the integral can take. By interpreting this as a hemisphere, the transition to spherical coordinates becomes natural. For our integration, we use \( \rho \) from 0 to \( \sqrt{90} \) (the radius), \( \phi \) from 0 to \( \frac{\pi}{2} \) (ensuring \( z \geq 0 \)), and \( \theta \) from 0 to \( 2\pi \) (to cover the complete circular rotation on the xy-plane). This gives us a complete understanding of the space through which we are integrating.

When integrating in different coordinate systems, such as moving from Cartesian to spherical coordinates, adjustments must be made to account for changes in the volume element size. This is where the Jacobian transformation comes into play.
The Jacobian is a determinant used in changing variables to account for how the coordinate system skews or compresses the volume. In spherical coordinates, the Jacobian transformation factor is \( \rho^2 \sin \phi \).
This ensures the volume element \( dV \) in Cartesian coordinates \( (dx \, dy \, dz) \) is correctly represented in spherical coordinates as \( (\rho^2 \sin \phi \, d\rho \, d\phi \, d\theta) \).
The Jacobian specifically does the following:

• It scales the volume element correctly to match the geometry of the system.
• Accounts for stretching and skewing that occurs when transforming to curvilinear coordinates.

It is essential to apply this transformation to solve triple integrals correctly when transitioning from Cartesian to spherical coordinates, ensuring an accurate calculation of volume or mass within a defined region.