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Find the area bounded by the following curves and lines.

To find the area bounded by the curves \( y = 2\sin x \) and \( y = \sin 2x \) over the interval \( 0 \leq x \leq \pi \), we follow these steps:

### 1. Set up the area integral:

The area between two curves is given by the integral:

\[
\text{Area} = \int_a^b \left( f(x) - g(x) \right) dx
\]

Here, \( f(x) = 2\sin x \) (the upper curve) and \( g(x) = \sin 2x \) (the lower curve). The limits of integration are from \( 0 \) to \( \pi \).

Thus, the area is:

\[
\text{Area} = \int_0^\pi \left( 2\sin x - \sin 2x \right) dx
\]

### 2. Compute the integral:

We now compute the integral of each function separately:

\[
\int_0^\pi 2\sin x \, dx = 2 \left[ -\cos x \right]_0^\pi = 2 \left( -(-1) - (-1) \right) = 2 \times 2 = 4
\]

For the second term:

\[
\int_0^\pi \sin 2x \, dx = \frac{-1}{2} \left[ \cos 2x \right]_0^\pi = \frac{-1}{2} \left( \cos 2\pi - \cos 0 \right) = \frac{-1}{2} (1 - 1) = 0
\]

### 3. Final answer:

The area is:

\[
\text{Area} = 4 - 0 = 4
\]

Thus, the area bounded by the curves \( y = 2\sin x \) and \( y = \sin 2x \) between \( 0 \) and \( \pi \) is \( \boxed{4} \).

---

Would you like more details on any step? Here are some related questions for further practice:

1. What happens to the area if we integrate over a different interval?
2. How does the area change if the upper curve is modified to \( y = 3\sin x \)?
3. What is the area between \( y = \cos x \) and \( y = \sin x \) over the interval \( 0 \leq x \leq \pi/2 \)?
4. How would you find the points of intersection between \( y = 2\sin x \) and \( y = \sin 2x \)?
5. Can you generalize this method for any pair of sinusoidal functions?

**Tip**: When finding the area between curves, always make sure to carefully identify which curve is on top over the entire interval to avoid negative areas.

Find the area bounded by the curves y = 2sin(x) and y = sin(2x) over the interval 0 ≤ x ≤ π.

Learn how to find the area bounded by the curves y = 2sin(x) and y = sin(2x) over the interval 0 ≤ x ≤ π. This problem involves key calculus concepts like integration of trigonometric functions and the formula for the area between curves.

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