Triple Integral in Spherical Coordinates2

Integrate the region described in spherical coordinates where $\rho = 1$, $0 \leq \phi \leq \frac{\pi}{6}$, and $0 \leq \theta \leq \frac{3\pi}{2}$.

In this problem, you are tasked with finding the volume of a region bounded by specific limits in spherical coordinates. Spherical coordinates provide an alternative to the traditional Cartesian coordinates for dealing with problems involving symmetry around a point, often simplifying integrals in three-dimensional space. This method is particularly advantageous when dealing with spherical or radial symmetry because it aligns neatly with the geometry of the problem.

To solve problems like this one, first visualize the region of integration in terms of spheres or shells defined by rho, phi, and theta. Understand that rho represents the radius from the origin, phi is the angle from the positive z-axis, and theta is the angle in the xy-plane from the positive x-axis. The given conditions restrict the region to a particular segment of a sphere, making it essential to properly interpret these bounds to set up the integral correctly.

When integrating in spherical coordinates, remember to include the Jacobian determinant, which for spherical coordinates is rho squared multiplied by the sine of phi. This accounts for the change in volume element from Cartesian to spherical coordinates. The problem here has a somewhat straightforward integration path due to the fixed radius and symmetric angular bounds, but visualizing the solid and correctly applying the spherical coordinate transformations is crucial to finding the correct solution.