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Use a calculator to approximate the area. The net​ area, approximated using the left Riemann sum with nequals​4, is

The image you've provided shows a problem involving the approximation of the area using the left Riemann sum. Let's break down the steps required to calculate the left Riemann sum for \( f(x) = \sin(3x) \) over the interval \( \left[ \frac{\pi}{2}, \pi \right] \) with \( n = 4 \).

### Step-by-Step Approach:

1. **Determine the Subintervals:**
   \[
   \Delta x = \frac{\pi - \frac{\pi}{2}}{4} = \frac{\pi}{2} \times \frac{1}{4} = \frac{\pi}{8}
   \]
   The subintervals will be:
   \[
   x_0 = \frac{\pi}{2}, \quad x_1 = \frac{\pi}{2} + \frac{\pi}{8}, \quad x_2 = \frac{\pi}{2} + 2\frac{\pi}{8}, \quad x_3 = \frac{\pi}{2} + 3\frac{\pi}{8}, \quad x_4 = \pi
   \]

2. **Evaluate the function \( f(x) = \sin(3x) \) at each left endpoint \( x_0, x_1, x_2, x_3 \):**
   - \( f(x_0) = \sin\left(3 \times \frac{\pi}{2}\right) \)
   - \( f(x_1) = \sin\left(3 \times \left(\frac{\pi}{2} + \frac{\pi}{8}\right)\right) \)
   - \( f(x_2) = \sin\left(3 \times \left(\frac{\pi}{2} + 2\frac{\pi}{8}\right)\right) \)
   - \( f(x_3) = \sin\left(3 \times \left(\frac{\pi}{2} + 3\frac{\pi}{8}\right)\right) \)

3. **Calculate the Left Riemann Sum:**
   The left Riemann sum is given by:
   \[
   L_4 = \Delta x \left[ f(x_0) + f(x_1) + f(x_2) + f(x_3) \right]
   \]

I'll calculate these values now.The left Riemann sum approximation for the net area using \( n = 4 \) is approximately:

\[
L_4 \approx 0.098
\]

This value is rounded to three decimal places, as required. Let me know if you'd like further details or explanations!

Here are some related questions for deeper understanding:

1. How does the left Riemann sum compare with the right and midpoint Riemann sums?
2. What happens to the approximation as the number of subintervals \( n \) increases?
3. Can you explain why we use \(\sin(3x)\) and how it affects the behavior of the graph?
4. What would be the effect of using different methods like the trapezoidal rule or Simpson's rule?
5. How does the negative and positive area in this interval contribute to the net area?

**Tip**: Riemann sums provide a foundation for understanding definite integrals and area under curves.

Use a calculator to approximate the area. The net​ area, approximated using the left Riemann sum with n=4, is?

Use a calculator to approximate the net area using a left Riemann sum with n=4 for the function f(x) = sin(3x) on the interval [π/2, π]. This problem involves key concepts in calculus such as approximation of areas under curves and Riemann sums.

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