Answer
The surface area of the ellipsoid obtained by rotating the given ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ about the x-axis or y-axis is $S = 2\pi * \left(a^2 - b^2 + \frac{ab}{\sqrt{a^2 - b^2}} * E\left(\frac{a^2 - b^2}{a^2}\right)\right)$, where $E(e^2)$ is the complete elliptic integral of the second kind.
Work Step by Step
An ellipsoid is a surface that can be obtained by rotating an ellipse about one of its principal axes. In this case, the given ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is rotated about the x-axis to obtain an ellipsoid.
The surface area of an ellipsoid can be calculated using the formula: $S = 2\pi * \left(c^2 + \frac{ab}{e} * E(e^2)\right)$, where $a$ and $b$ are the semi-major and semi-minor axes of the ellipse, respectively, $c$ is the distance from the center of the ellipse to one of its foci, $e$ is the eccentricity of the ellipse, and $E(e^2)$ is the complete elliptic integral of the second kind.
In this case, $a$ and $b$ are given by the equation of the ellipse. The distance from the center of the ellipse to one of its foci is given by $c = \sqrt{a^2 - b^2}$. The eccentricity of the ellipse is given by $e = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}$. Substituting these values into the formula for the surface area of an ellipsoid, we get:
$$S = 2\pi * \left(c^2 + \frac{ab}{e} * E(e^2)\right)$$
$$= 2\pi * \left(a^2 - b^2 + \frac{ab}{\sqrt{a^2 - b^2}} * E\left(\frac{a^2 - b^2}{a^2}\right)\right).$$
This formula gives us the surface area of the ellipsoid obtained by rotating the given ellipse about the x-axis. However, to obtain a numerical value for the surface area, we need to know the values of $a$ and $b$.
