To evaluate the given double integral, let's go through the problem step-by-step.

The integral is written in polar coordinates as:

$\int_0^{\pi} \int_0^5 r \cos(\theta) \, r \, dr \, d\theta$

The function $f(r, \theta) = r \cos(\theta)$ is multiplied by $r$ due to the Jacobian determinant when converting to polar coordinates.

Step 1: Simplify the Integrand

The integrand becomes: $r \cdot r \cos(\theta) = r^2 \cos(\theta)$

Thus, the integral can be rewritten as: $\int_0^{\pi} \int_0^5 r^2 \cos(\theta) \, dr \, d\theta$

Step 2: Integrate with Respect to $r$

Now, we focus on the inner integral with respect to $r$: $\int_0^{\pi} \left( \int_0^5 r^2 \cos(\theta) \, dr \right) d\theta$

Since $\cos(\theta)$ is independent of $r$, we can factor it out: $= \int_0^{\pi} \cos(\theta) \left( \int_0^5 r^2 \, dr \right) d\theta$

Evaluate the inner integral: $\int_0^5 r^2 \, dr = \left[ \frac{r^3}{3} \right]_0^5 = \frac{5^3}{3} = \frac{125}{3}$

Thus, the integral becomes: $= \int_0^{\pi} \cos(\theta) \cdot \frac{125}{3} \, d\theta = \frac{125}{3} \int_0^{\pi} \cos(\theta) \, d\theta$

Step 3: Integrate with Respect to $\theta$

Now, evaluate the remaining integral: $\int_0^{\pi} \cos(\theta) \, d\theta$

The integral of $\cos(\theta)$ is $\sin(\theta)$: $\int_0^{\pi} \cos(\theta) \, d\theta = \left[ \sin(\theta) \right]_0^{\pi} = \sin(\pi) - \sin(0) = 0 - 0 = 0$

Final Answer

Since this last integral evaluates to zero, the value of the entire double integral is: $\int_0^{\pi} \int_0^5 r^2 \cos(\theta) \, dr \, d\theta = 0$

Region $R$

The region $R$ in polar coordinates is defined by:

• $r$ ranging from 0 to 5 (a radius up to 5 units)
• $\theta$ ranging from 0 to $\pi$ (a half-circle in the first and second quadrants)

So, $R$ represents the upper half of a disk with radius 5.

Would you like a more detailed breakdown of any part of this solution, or do you have questions about the region sketch?

Related Questions:

1. What is the significance of using polar coordinates in double integrals?
2. How would the integral change if the limits for $\theta$ were from $0$ to $2\pi$?
3. What is the purpose of the Jacobian in coordinate transformations?
4. How do we determine the appropriate region $R$ in polar coordinates?
5. Can we convert any double integral from Cartesian to polar coordinates?

Tip: When integrating in polar coordinates, always remember to include $r$ in the integrand to account for the area element $dA = r \, dr \, d\theta$.