The image you've provided is related to finding the flux of the vector field $\mathbf{F}(x, y, z) = (3xy^2, 3x^2y, z^3)$ out of a sphere of radius 8 centered at the origin, using spherical coordinates.

From the setup, you’re given the integral for flux in spherical coordinates: $\int_{0}^{\pi} \int_{0}^{2\pi} \int_{0}^{8} f(\theta, \phi) \, \rho^2 \sin \theta \, d\rho \, d\theta \, d\phi$ You need to find the value of the integral.

The flux value given in the image is $78643.2$, and it appears the limits of integration for $\theta$ and $\phi$ are set correctly as $\theta$ ranges from 0 to $\pi$, and $\phi$ ranges from 0 to $2\pi$, with $\rho$ running from 0 to 8 (the radius of the sphere).

If you want to go over any further details about spherical coordinates, flux calculation, or work through any part of the process, feel free to ask!

Would you like to discuss this step by step?