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Your document contains problems on **approximating areas with Riemann sums** and **trapezoidal approximations**, along with some applications to real-world scenarios like weight loss, balloon height, and particle movement.

Here’s a breakdown of the problems and solutions:

1. **Midpoint Riemann Sum (n = 3) for \( f(x) = x^2 + x + 7 \) on [1,2]**  
   **Solution:** 20.175

2. **Left Riemann Sum (n = 5) for \( f(x) = 6x + 5 \) on [-2,2]**  
   **Solution:** 10.4

3. **Trapezoidal Approximation (n = 4) for \( f(x) = 0.2x^3 - x + 12 \) on [1,3]**  
   **Solution:** 42

4. **Weight loss per week (using Left Riemann Sum)**  
   **a. Computation result:** 124  
   **b. Interpretation:** Total pounds lost from week 2 to week 11.

5. **Hot air balloon height (using Trapezoidal Approximation)**  
   **a. Computation result:** 75.2  
   **b. Height at 11 minutes:** 125.2 meters

6. **Particle movement (using Midpoint Riemann Sum with n=4)**  
   **a. Computation result:** 416  
   **b. Interpretation:** Total distance traveled from 0 to 16 seconds.

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Would you like a step-by-step explanation for any of these solutions?  

Here are some related questions to deepen understanding:
1. What is the difference between Midpoint, Left, and Right Riemann sums?
2. How does increasing the number of subintervals (n) improve the approximation?
3. When is the Trapezoidal Rule a better choice than Riemann sums?
4. How do Riemann sums approximate definite integrals?
5. Can we use Simpson's Rule for better accuracy? How does it compare?

**Tip:** **Midpoint Riemann sums** often provide a better approximation than left or right sums because they balance overestimations and underestimations more effectively.

Your document contains problems on approximating areas with Riemann sums and trapezoidal approximations, along with some applications to real-world scenarios like weight loss, balloon height, and particle movement.

Explore problems on approximating areas using Riemann sums and trapezoidal approximations, including applications to real-world scenarios such as weight loss, balloon height, and particle movement. Learn key methods like the Midpoint Riemann Sum, Left Riemann Sum, and Trapezoidal Rule.

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