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.

---

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.