The given problem is to evaluate the triple integral:

$\iiint_V dz\, dy\, dx$

over the volume of a sphere $x^2 + y^2 + z^2 = a^2$.

Step 1: Convert to spherical coordinates

The equation of the sphere $x^2 + y^2 + z^2 = a^2$ suggests that spherical coordinates are the most suitable for this problem. The relations between Cartesian and spherical coordinates are as follows:

$x = r \sin \theta \cos \phi, \quad y = r \sin \theta \sin \phi, \quad z = r \cos \theta$

where:

• $r$ is the radial distance,
• $\theta$ is the polar angle (measured from the positive $z$-axis),
• $\phi$ is the azimuthal angle (measured from the positive $x$-axis).

The volume element in spherical coordinates is:

$dV = r^2 \sin \theta \, dr \, d\theta \, d\phi$

Step 2: Set the limits of integration

For a full sphere, the limits of integration are:

• $r$ varies from $0$ to $a$,
• $\theta$ varies from $0$ to $\pi$,
• $\phi$ varies from $0$ to $2\pi$.

Step 3: Write the integral in spherical coordinates

The triple integral over the volume of the sphere in spherical coordinates becomes:

$\iiint_V dz\, dy\, dx = \int_0^{2\pi} \int_0^\pi \int_0^a r^2 \sin \theta \, dr \, d\theta \, d\phi$

Step 4: Evaluate the integral

1. Integrate with respect to $r$:

$\int_0^a r^2 \, dr = \left[ \frac{r^3}{3} \right]_0^a = \frac{a^3}{3}$

2. Integrate with respect to $\theta$:

$\int_0^\pi \sin \theta \, d\theta = \left[ -\cos \theta \right]_0^\pi = -(-1 - 1) = 2$

3. Integrate with respect to $\phi$:

$\int_0^{2\pi} d\phi = 2\pi$

Step 5: Combine the results

Now, multiply the results of each step:

$\frac{a^3}{3} \times 2 \times 2\pi = \frac{4\pi a^3}{3}$

Final Answer:

The value of the triple integral over the volume of the sphere is:

$\iiint_V dz\, dy\, dx = \frac{4\pi a^3}{3}$

This is the volume of the sphere, which makes sense since the integrand is simply 1 (as we were integrating $dz \, dy \, dx$, which is the volume element in Cartesian coordinates).

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Do you want more details or have any further questions?

Here are 5 related questions:

1. How would you evaluate the same integral over a hemisphere?
2. What happens if the limits on $\theta$ are different, e.g., from $0$ to $\frac{\pi}{2}$?
3. How would the integral change if the integrand was $z$ instead of 1?
4. Can you explain how to derive the volume element $r^2 \sin \theta \, dr \, d\theta \, d\phi$ in spherical coordinates?
5. How would you calculate the surface area of the sphere using spherical coordinates?

Tip: When solving integrals over regions with symmetry, such as spheres, converting to spherical or cylindrical coordinates simplifies the process significantly.