The given problem asks to compute the double integral:

$\int_{4}^{7} \int_{\pi}^{2\pi} r \, d\theta \, dr$

This is a double integral in polar coordinates where the integrand is $r$. The integral is split into two parts, with limits for $r$ ranging from 4 to 7 and $\theta$ ranging from $\pi$ to $2\pi$.

Step-by-step solution:

1. Inner Integral: The inner integral is with respect to $\theta$, treating $r$ as a constant: $\int_{\pi}^{2\pi} r \, d\theta = r \left[ \theta \right]_{\pi}^{2\pi} = r \cdot (2\pi - \pi) = r \cdot \pi$ So, the result of the inner integral is $r\pi$.

2. Outer Integral: Now, integrate the result $r\pi$ with respect to $r$: $\int_{4}^{7} r\pi \, dr = \pi \int_{4}^{7} r \, dr = \pi \left[ \frac{r^2}{2} \right]_{4}^{7}$ $= \pi \left( \frac{7^2}{2} - \frac{4^2}{2} \right) = \pi \left( \frac{49}{2} - \frac{16}{2} \right)$ $= \pi \left( \frac{33}{2} \right) = \frac{33\pi}{2}$

Final Answer:

The value of the double integral is $\frac{33\pi}{2}$.

Would you like further details or have any questions? Here are some related questions to extend your understanding:

1. How do you change a Cartesian integral into a polar form?
2. What are the applications of double integrals in polar coordinates?
3. How would the limits change if the problem was set in Cartesian coordinates?
4. What is the geometric interpretation of this double integral?
5. How would the result differ if the limits for $\theta$ were from 0 to $\pi$?

Tip: Always check the order of integration and ensure that each variable is integrated within its correct limits!