To solve the given problem, we need to compute the volume of the ellipsoid given by:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$

Step 1: Volume formula for an ellipsoid

The volume of an ellipsoid is calculated using a standard formula derived from multivariable calculus:

$V = \frac{4}{3} \pi a b c$

Where $a$, $b$, and $c$ are the semi-axes of the ellipsoid.

Step 2: Integration Derivation (Optional Explanation)

To derive this, we use a change of variables that scales the ellipsoid into a unit sphere: Let: $u = \frac{x}{a}, \quad v = \frac{y}{b}, \quad w = \frac{z}{c}.$ Under this transformation, the ellipsoid becomes the sphere: $u^2 + v^2 + w^2 = 1.$

The Jacobian determinant for this transformation is $abc$, so the volume of the ellipsoid is scaled by this factor relative to the volume of the unit sphere.

The volume of the unit sphere is $\frac{4}{3} \pi$. Multiplying by the scaling factor $abc$ gives: $V = abc \cdot \frac{4}{3} \pi = \frac{4}{3} \pi a b c.$

Step 3: Final Answer

The volume of the ellipsoid is:

$V = \frac{4}{3} \pi a b c$

Would you like further clarification or details on how this formula was derived? Let me know!

Related Questions:

1. How is the Jacobian determinant used in multivariable calculus to change variables?
2. Why does the unit sphere have a volume of $\frac{4}{3} \pi$?
3. Can you calculate the surface area of an ellipsoid?
4. What happens if $a = b = c$? How does the ellipsoid become a sphere?
5. How would the volume change if one axis of the ellipsoid approaches zero?

Tip:

For ellipsoids with symmetry along axes, always verify the semi-axes lengths ($a$, $b$, and $c$) before applying the formula!