To solve the problem, we will approach the three tasks step by step:

1. Calculate the Volume of the Egg

The egg can be modeled as an ellipsoid, where the volume is given by:

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

Here:

• $a$ is the semi-major axis of the egg (half of the length along the x-axis),
• $b$ is the semi-minor axis (half of the width along the y-axis), and
• $c$ can be considered the semi-axis along the z-axis.

However, you mentioned using the integral formula for volume: $V_x = \pi \int_a^b f(x)^2 \, dx$

This can be interpreted as finding the volume of revolution for the egg-shaped profile $f(x)$ (the cross-sectional area), rotated around the x-axis. For an ellipsoid, the general equation is:

$\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1$

Solving for $y = f(x)$ gives: $f(x) = b \sqrt{1 - \left(\frac{x}{a}\right)^2}$

Thus, the volume integral is: $V_x = \pi \int_{-a}^{a} \left(b \sqrt{1 - \frac{x^2}{a^2}}\right)^2 dx = \pi b^2 \int_{-a}^{a} \left(1 - \frac{x^2}{a^2}\right) dx$

Evaluating this integral: $V_x = \pi b^2 \left[ x - \frac{x^3}{3a^2} \right]_{-a}^{a}$

At $x = a$ and $x = -a$, the terms simplify to:

= \pi b^2 \left(2a - \frac{2a}{3}\right) = \pi b^2 \cdot \frac{4a}{3}$$ Thus, the volume becomes: $$V_x = \frac{4}{3} \pi a b^2$$ Substitute $$a = 2.9$$ and $$b = 2.1$$: $$V_x = \frac{4}{3} \pi (2.9) (2.1)^2$$ $$V_x \approx \frac{4}{3} \pi (2.9) (4.41)$$ $$V_x \approx \frac{4}{3} \pi (12.789)$$ $$V_x \approx 53.625 \, \text{cm}^3$$ ### 2. **Calculate the Periphery (Circumference)** The formula for calculating the periphery or arc length of the egg's curve is: $$L = \int_{-a}^{a} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$ For the egg-shaped ellipsoid, $$y = b \sqrt{1 - \frac{x^2}{a^2}}$$, and the derivative $$\frac{dy}{dx}$$ is: $$\frac{dy}{dx} = \frac{-b^2 x}{a^2 \sqrt{1 - \frac{x^2}{a^2}}}$$ Substitute this into the arc length formula: $$L = \int_{-a}^{a} \sqrt{1 + \left( \frac{-b^2 x}{a^2 \sqrt{1 - \frac{x^2}{a^2}}} \right)^2} \, dx$$ This integral is quite complicated analytically, so it would typically be evaluated numerically. ### 3. **Calculate the Mass of the Eggshell** The volume $$V$$ has been calculated already. To calculate the mass of the eggshell, we can use the formula: $$\text{Mass} = V \cdot \rho$$ where: - $$\rho$$ is the density of the eggshell (mass per unit volume), - $$V$$ is the volume of the eggshell. If the eggshell thickness is $$h$$, and you have the overall volume of the egg, the mass is: $$\text{Mass} = V \cdot h \cdot \rho$$ You can calculate this if the density $$\rho$$ and thickness $$h$$ are given. --- Would you like more details on any of these steps or assistance with the numerical integration for the periphery? Here are 5 related questions for further exploration: 1. How would the volume change if the semi-minor axis was altered? 2. How would the thickness of the eggshell affect the total mass? 3. How can the volume integral be generalized for other shapes? 4. What numerical methods can be used to solve the arc length integral? 5. How would the formula for volume differ for a prolate spheroid? **Tip**: Numerical methods like Simpson's rule or Gaussian quadrature can be very helpful for solving complex integrals.