The integral provided needs to be evaluated by transforming the given rectangular coordinates $(x, y, z)$ into spherical coordinates. Here are the steps for solving this:

Step 1: Recall spherical coordinates definitions

In spherical coordinates: $x = \rho \sin\phi \cos\theta, \quad y = \rho \sin\phi \sin\theta, \quad z = \rho \cos\phi$ where:

• $\rho$ is the radial distance,
• $\phi$ is the angle between the positive $z$-axis and the line connecting the origin to the point (polar angle),
• $\theta$ is the angle between the positive $x$-axis and the projection of the point onto the $xy$-plane (azimuthal angle).

The Jacobian for the transformation to spherical coordinates is: $dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$

Step 2: Convert bounds

The given bounds describe a region in a quarter of a sphere:

1. $x^2 + y^2 + z^2 \leq 16$: The radius of the sphere is 4 ($\rho \leq 4$).
2. $x \geq 0$, $y \geq 0$: The integration is limited to the first octant ($0 \leq \theta \leq \frac{\pi}{2}$, $0 \leq \phi \leq \frac{\pi}{2}$).

Step 3: Rewrite the integrand

The integrand $xy$ in spherical coordinates becomes: $xy = (\rho \sin\phi \cos\theta)(\rho \sin\phi \sin\theta) = \rho^2 \sin^2\phi \cos\theta \sin\theta$

Step 4: Set up the integral

The triple integral in spherical coordinates is: $\int_0^{\pi/2} \int_0^{\pi/2} \int_0^4 \rho^2 \sin^2\phi \cos\theta \sin\theta \cdot \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$ Simplify the integrand: $\rho^2 \cdot \rho^2 = \rho^4, \quad \sin^2\phi \cdot \sin\phi = \sin^3\phi, \quad \cos\theta \sin\theta$ So, the integral becomes: $\int_0^{\pi/2} \int_0^{\pi/2} \int_0^4 \rho^4 \sin^3\phi \cos\theta \sin\theta \, d\rho \, d\phi \, d\theta$

Step 5: Solve the integral

1. Integrate with respect to $\rho$: $\int_0^4 \rho^4 \, d\rho = \left[\frac{\rho^5}{5}\right]_0^4 = \frac{4^5}{5} = \frac{1024}{5}$
2. Integrate with respect to $\theta$: $\int_0^{\pi/2} \cos\theta \sin\theta \, d\theta = \int_0^{\pi/2} \frac{1}{2} \sin(2\theta) \, d\theta = \frac{1}{2} \left[-\frac{1}{2}\cos(2\theta)\right]_0^{\pi/2} = \frac{1}{2} \cdot \frac{1}{2} \cdot (1 - (-1)) = \frac{1}{2}$
3. Integrate with respect to $\phi$: $\int_0^{\pi/2} \sin^3\phi \, d\phi = \int_0^{\pi/2} \sin\phi (1 - \cos^2\phi) \, d\phi = \int_0^{\pi/2} \sin\phi \, d\phi - \int_0^{\pi/2} \sin\phi \cos^2\phi \, d\phi$ For the first term: $\int_0^{\pi/2} \sin\phi \, d\phi = [-\cos\phi]_0^{\pi/2} = 1$ For the second term, use substitution $u = \cos\phi$, $du = -\sin\phi \, d\phi$: $\int_0^{\pi/2} \sin\phi \cos^2\phi \, d\phi = -\int_1^0 u^2 \, du = \int_0^1 u^2 \, du = \left[\frac{u^3}{3}\right]_0^1 = \frac{1}{3}$ So: $\int_0^{\pi/2} \sin^3\phi \, d\phi = 1 - \frac{1}{3} = \frac{2}{3}$

Step 6: Combine the results

The value of the integral is: $\frac{1024}{5} \cdot \frac{1}{2} \cdot \frac{2}{3} = \frac{1024}{15}$

The final result is: $\boxed{\frac{1024}{15}}$

Let me know if you want further details or have any questions.
Here are some related questions you might explore:

1. How are spherical coordinates derived from Cartesian coordinates?
2. What is the significance of the Jacobian in coordinate transformations?
3. How would this integral change if integrated over the entire sphere?
4. How is $\sin^3\phi$ integrated using substitution?
5. What are the applications of spherical coordinates in physics?

Tip: Always visualize the bounds of integration to verify the region described by the limits.