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To find the area of a disk of radius 11 using a line integral, we will use Green's Theorem. For Green's Theorem, we need to convert the double integral over the region into a line integral over the boundary.

The parameterization for a circle of radius \( R \) is given by:
\[ x = R \cos t \]
\[ y = R \sin t \]
where \( R = 11 \) and \( t \) ranges from \( 0 \) to \( 2\pi \).

Green's Theorem states:
\[ \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA = \oint_C \left( P \, dx + Q \, dy \right) \]

To find the area of the disk, we can choose \( P = 0 \) and \( Q = x \). Then,
\[ \frac{\partial Q}{\partial x} = 1 \]
\[ \frac{\partial P}{\partial y} = 0 \]
So, the area is:
\[ \text{Area} = \iint_R 1 \, dA = \oint_C x \, dy \]

Using the parameterization:
\[ x = 11 \cos t \]
\[ y = 11 \sin t \]
\[ dy = 11 \cos t \, dt \]

The line integral becomes:
\[ \oint_C x \, dy = \int_0^{2\pi} (11 \cos t) \cdot (11 \cos t \, dt) = \int_0^{2\pi} 121 \cos^2 t \, dt \]

To simplify this integral, we use the trigonometric identity:
\[ \cos^2 t = \frac{1 + \cos 2t}{2} \]

Thus,
\[ \int_0^{2\pi} 121 \cos^2 t \, dt = 121 \int_0^{2\pi} \frac{1 + \cos 2t}{2} \, dt = 121 \left( \int_0^{2\pi} \frac{1}{2} \, dt + \int_0^{2\pi} \frac{\cos 2t}{2} \, dt \right) \]

The first integral evaluates to:
\[ \int_0^{2\pi} \frac{1}{2} \, dt = \frac{1}{2} \times 2\pi = \pi \]

The second integral evaluates to zero because \( \cos 2t \) over a full period is zero:
\[ \int_0^{2\pi} \frac{\cos 2t}{2} \, dt = 0 \]

Thus,
\[ \int_0^{2\pi} 121 \cos^2 t \, dt = 121 \times \pi = 121\pi \]

So the area of the disk of radius 11 is \( 121\pi \).

The integral in the provided problem can be set up as follows:
\[ \int_0^{11} 121 \, dt = 121t \Big|_0^{11} = 121 \cdot 11 = 1331 \]

However, since this doesn't match the disk's area, re-checking initial boundary conditions is crucial. 

Thus, the integral for area based on disk radius should follow standard area calculation methods using circle integrals for validation:
\[ 11\pi \]

Please clarify if further, detailed information or explanations are required.

Here are 5 questions you might find useful:
1. How does Green's Theorem apply to finding the area of a disk?
2. Can you explain the parameterization of a circle?
3. Why do we use the trigonometric identity \(\cos^2 t = \frac{1 + \cos 2t}{2}\) in integration?
4. What is the significance of setting \(P = 0\) and \(Q = x\) in Green's Theorem for this problem?
5. How would you solve similar problems for shapes other than a disk?

**Tip:** When using Green's Theorem, always ensure the chosen \(P\) and \(Q\) functions simplify the problem effectively.

Explore how to use Green's Theorem to calculate the area of a disk with a radius of 11. Learn about line integrals, parameterization of curves, and trigonometric identities used in this mathematical problem.

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