To estimate the area under the curve of $f(x) = 2\sin(x)$ from $x = 0$ to $x = \pi$ using the midpoints rule with two rectangles, follow these steps:

Step 1: Divide the interval

The interval of interest is $[0, \pi]$. Since we are using two rectangles, divide this interval into two subintervals:

$\Delta x = \frac{\pi - 0}{2} = \frac{\pi}{2}$

So, the two subintervals are:

1. $[0, \frac{\pi}{2}]$
2. $[\frac{\pi}{2}, \pi]$

Step 2: Find the midpoints of the subintervals

The midpoints of the two subintervals are:

1. Midpoint of $[0, \frac{\pi}{2}]$ is $x_1 = \frac{\pi}{4}$
2. Midpoint of $[\frac{\pi}{2}, \pi]$ is $x_2 = \frac{3\pi}{4}$

Step 3: Evaluate the function at the midpoints

Now, evaluate $f(x) = 2\sin(x)$ at these midpoints:

$f(x_1) = f\left(\frac{\pi}{4}\right) = 2\sin\left(\frac{\pi}{4}\right) = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}$

$f(x_2) = f\left(\frac{3\pi}{4}\right) = 2\sin\left(\frac{3\pi}{4}\right) = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}$

Step 4: Approximate the area using the midpoints rule

The area of each rectangle is given by the height of the function at the midpoint times the width of the subinterval:

$\text{Area} \approx f(x_1) \cdot \Delta x + f(x_2) \cdot \Delta x$

Substitute the values:

$\text{Area} \approx \sqrt{2} \cdot \frac{\pi}{2} + \sqrt{2} \cdot \frac{\pi}{2} = 2\sqrt{2} \cdot \frac{\pi}{2} = \pi\sqrt{2}$

Thus, the estimated area under the curve is:

$\boxed{\pi\sqrt{2}}$

---

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. How would the estimate change if we used 4 rectangles instead of 2?
2. What is the actual area under the curve $f(x) = 2\sin(x)$ from $0$ to $\pi$?
3. Can you explain the difference between the midpoint rule and the trapezoidal rule?
4. How does increasing the number of rectangles affect the accuracy of the midpoint rule?
5. What is the error bound for the midpoint rule in this case?

Tip: The more rectangles you use in a Riemann sum or midpoint approximation, the closer the estimate will be to the actual area under the curve!