The image you've provided shows a vector calculus problem involving surface integrals. The goal seems to be calculating the integral $I$, which is presented as:

$I = \iint_{\Sigma} \frac{x \, dy \, dz + y \, dz \, dx + z \, dx \, dy}{(x^2 + y^2 + z^2)^{3/2}}$

This represents a surface integral over a surface $\Sigma$, with two different surfaces provided for the calculation:

1. Surface 1: $\Sigma$ is the upper part of the ellipsoid defined by $z = \sqrt{a^2 - x^2 - y^2}$.

2. Surface 2: $\Sigma$ is the upper hemisphere of an ellipsoid given by $\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{25} = 1$, with $z \geq 0$.

This integral resembles one for computing the solid angle subtended by a surface at the origin, possibly based on the divergence theorem or Gauss’s law for gravitation/electric fields.

Approach to the solution:

The term inside the integral suggests that this is related to calculating the flux of a vector field $\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^3}$ over the surface $\Sigma$. This form arises in problems dealing with inverse-square laws in physics, such as gravitational or electric fields.

Steps:

1. Use of the Divergence Theorem (Gauss's Theorem): The vector field $\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^3}$, where $\mathbf{r} = (x, y, z)$, has a divergence $\nabla \cdot \mathbf{F} = 0$ everywhere except at the origin. This suggests the integral could be simplified using symmetry arguments or known results.

2. For Surface 1 (Ellipsoid upper cap): The surface is symmetric, and because it only covers part of the full ellipsoid, we can use the symmetry properties of ellipsoids and spherical coordinates to evaluate the integral.

3. For Surface 2 (Upper hemisphere of the ellipsoid): Similarly, this is a portion of an ellipsoid that can be treated using ellipsoidal coordinates, and known results about surface integrals over ellipsoidal surfaces might help.

Further Steps:

• Apply the proper parametrization for both surfaces and compute the integral directly or use known results for these types of integrals.

Would you like a more detailed explanation of each surface's integral, or clarification on a specific step? Let me know if you want details on the calculation for either surface.

Here are some related questions:

1. How do you parametrize the surface of an ellipsoid for integral calculations?
2. What role does the divergence theorem play in simplifying surface integrals of vector fields?
3. How would the integral change if we considered the full ellipsoid instead of the upper part?
4. What physical meaning does this type of surface integral have in terms of fields like gravitation or electrostatics?
5. How can symmetry be exploited in simplifying surface integrals?

Tip: Always check if the problem exhibits symmetry properties, as this can significantly simplify the integral and reduce computation time.