To find the mass of the rod, we integrate the density function $p(x)$ over the given interval. The formula for the mass of a thin rod is:

$M = \int_{a}^{b} p(x) \, dx$

where $p(x)$ is the density function and $[a, b]$ is the interval along the x-axis.

In this case:

• $p(x) = 4x^2 + 2$
• The rod extends over the interval $[5, 11]$

So, the mass $M$ is:

$M = \int_{5}^{11} (4x^2 + 2) \, dx$

Let's compute this integral step by step:

1. Break down the integral: $M = \int_{5}^{11} 4x^2 \, dx + \int_{5}^{11} 2 \, dx$

2. Compute the integrals:

   • $\int 4x^2 \, dx = \frac{4}{3}x^3$
   • $\int 2 \, dx = 2x$

Now, apply the limits of integration.

For $\int_{5}^{11} 4x^2 \, dx$: $\left[\frac{4}{3}x^3\right]_{5}^{11} = \frac{4}{3}(11^3) - \frac{4}{3}(5^3)$

For $\int_{5}^{11} 2 \, dx$: $\left[2x\right]_{5}^{11} = 2(11) - 2(5)$

Let's calculate the exact value.The mass of the rod is $1620.0000$ units.

Would you like more details or have any questions about the process?

Here are five related questions to deepen your understanding:

1. How does the mass change if the interval is expanded to $[4, 12]$?
2. What happens if the density function is a linear function instead of quadratic?
3. Can you derive the center of mass for the rod based on this density function?
4. How would the result change if the density was $p(x) = 4x^3 + 2$ instead?
5. What are the physical interpretations of each term in the density function?

Tip: When integrating polynomial functions, split the integral into simpler components for easier calculations.