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It seems you’ve uploaded an image related to an integral problem. The instructions involve approximating an integral using left-hand sums (\( L_2 \)), right-hand sums (\( R_2 \)), midpoints (\( M_2 \)), and trapezoidal rules (\( T_2 \)). Let's break down the steps:

1. **\( L_2, R_2, M_2 \) Calculation:**
   - These are numerical methods for approximating the integral based on two subintervals. 
   - **\( L_2 \)**: Left-hand sum approximation using two rectangles.
   - **\( R_2 \)**: Right-hand sum approximation using two rectangles.
   - **\( M_2 \)**: Midpoint approximation using two midpoints for the subintervals.

   I'll need to analyze the graph of the function \( f(x) \) to help you compute these values. Since I cannot directly view the graph here, could you describe the graph or key points of the function over the interval \( [0, 4] \)? That way, I can guide you further.

2. **Estimations:**
   - Based on the shape of the function, we'll determine whether these estimates (left, right, and midpoint) are overestimates or underestimates.

3. **Trapezoidal Rule \( T_2 \)**: This is computed as the average of \( L_2 \) and \( R_2 \), providing a more refined approximation of the integral.

Please describe the graph or share more details so I can help you with these calculations. If you prefer, I can provide step-by-step guidance on how to find these values.

Would you like more details on how to approach each step? Let me know if you'd like assistance with any particular part of the problem!

Let I = ∫ from 0 to 4 of f(x) dx, where f is the function whose graph is shown below. (a) Use the graph to find L2, R2, and M2. (b) Are these underestimates or overestimates of I? (c) Use the graph to find T2.

Learn how to approximate the integral of a function f(x) from 0 to 4 using left-hand sums (L2), right-hand sums (R2), midpoints (M2), and trapezoidal rules (T2). Understand whether these methods provide underestimates or overestimates of the integral. Suitable for high school calculus students or early college learners.

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