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The problem provided involves finding the integral over an annular region \( R \), which lies between two circles defined by:

1. \( x^2 + y^2 = 1 \) (a circle with radius 1)
2. \( x^2 + y^2 = 5 \) (a circle with radius \( \sqrt{5} \))

The integral to evaluate is:

\[
\iint_R (x^2 + y^2) \, dA
\]

### Solution Approach:
We can simplify this problem by converting to polar coordinates since the region is circular (annular). In polar coordinates, the transformations are:

\[
x = r\cos\theta, \quad y = r\sin\theta, \quad \text{and} \quad x^2 + y^2 = r^2
\]

The area element \( dA \) in polar coordinates becomes \( r \, dr \, d\theta \).

Thus, the integral becomes:

\[
\iint_R (x^2 + y^2) \, dA = \int_{\theta=0}^{2\pi} \int_{r=1}^{\sqrt{5}} r^2 \cdot r \, dr \, d\theta
\]

This simplifies to:

\[
\int_0^{2\pi} \int_1^{\sqrt{5}} r^3 \, dr \, d\theta
\]

### Step 1: Evaluate the radial part of the integral
\[
\int_1^{\sqrt{5}} r^3 \, dr = \left[ \frac{r^4}{4} \right]_1^{\sqrt{5}} = \frac{(\sqrt{5})^4}{4} - \frac{1^4}{4} = \frac{25}{4} - \frac{1}{4} = \frac{24}{4} = 6
\]

### Step 2: Evaluate the angular part of the integral
\[
\int_0^{2\pi} d\theta = 2\pi
\]

### Step 3: Combine the results
The total integral is:

\[
6 \times 2\pi = 12\pi
\]

Thus, the value of the integral is \( \boxed{12\pi} \).

Would you like more details on any part of this solution or have further questions?

### Related Questions:
1. What would the integral be if the annular region were between \( x^2 + y^2 = 4 \) and \( x^2 + y^2 = 9 \)?
2. How does the area element change in polar coordinates for other geometric shapes, such as ellipses?
3. How would this integral change if we included a weight function like \( f(r,\theta) = r \sin(\theta) \)?
4. What if the integral had been \( \iint_R x \, dA \)? How would the approach differ?
5. How can polar coordinates simplify the calculation of integrals over circular regions in general?

### Tip:
When solving problems involving circular or radial symmetry, converting to polar coordinates often simplifies the calculation due to the natural geometry of the problem.

Let R be the annular region lying between the two circles x^2 + y^2 = 1 and x^2 + y^2 = 5. Evaluate the integral ∫∫(x^2 + y^2)dA.

Learn how to evaluate the double integral ∫∫(x^2 + y^2)dA over an annular region bounded by two circles using polar coordinates. The problem explores key multivariable calculus concepts such as area elements and the transformation to polar coordinates. Suitable for undergraduate students.

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