Consider a thin rod oriented along the x axis over the interval 8, 10. If the density of the rod is given by p(x)=5x^2 + 5, what is the mass of the rod rounded to 4 decimal places?


To find the mass of the rod, we need to integrate the density function \( p(x) = 5x^2 + 5 \) over the length of the rod, which is from \( x = 8 \) to \( x = 10 \).

The mass \( m \) of the rod can be computed using the formula:

\[
m = \int_{a}^{b} p(x) \, dx
\]

where \( p(x) \) is the density function, and \( a \) and \( b \) are the endpoints of the rod. Here, \( a = 8 \), \( b = 10 \), and \( p(x) = 5x^2 + 5 \).

Now let's compute this integral:

\[
m = \int_{8}^{10} (5x^2 + 5) \, dx
\]

First, break this integral into two parts:

\[
m = \int_{8}^{10} 5x^2 \, dx + \int_{8}^{10} 5 \, dx
\]

Now compute each integral separately:

1. \( \int 5x^2 \, dx = \frac{5x^3}{3} \)
2. \( \int 5 \, dx = 5x \)

Now, apply the limits of integration for both parts:

\[
m = \left[\frac{5x^3}{3}\right]_{8}^{10} + \left[5x\right]_{8}^{10}
\]

Substitute the limits of integration:

\[
m = \left(\frac{5(10)^3}{3} - \frac{5(8)^3}{3}\right) + \left(5(10) - 5(8)\right)
\]

Let's compute this step-by-step.

I'll calculate the values now.The mass of the rod, calculated by integrating the given density function, is approximately \( 823.3 \) units (rounded to 4 decimal places).

Would you like further clarification on this process?

Here are some related questions:
1. How would the mass change if the rod's length were different?
2. What happens if the density function is non-polynomial, like \( p(x) = e^x \)?
3. How would we compute the mass if the rod's density was variable in another direction (y or z)?
4. How does the mass relate to the moment of inertia for this rod?
5. What is the physical significance of integrating the density function?

**Tip:** Always ensure the density function is continuous over the interval to avoid complications during integration.

Consider a thin rod oriented along the x axis over the interval 8, 10. If the density of the rod is given by p(x)=5x^2 + 5, what is the mass of the rod rounded to 4 decimal places?

Calculate the mass of a thin rod along the x-axis between x = 8 and x = 10 using the density function p(x) = 5x^2 + 5. This step-by-step solution covers integration techniques and the application of the fundamental theorem of calculus. Suitable for high school and college-level students.

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