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This problem requires you to approximate the definite integral of the function \( f(x) = x^2 \sqrt{x^3 + 2} \) from \( x = 3 \) to \( x = 7 \), using the left endpoint, right endpoint, and midpoint rules for two different numbers of subintervals (four and eight rectangles). After calculating the approximations, you will compare them with the exact value of the integral and calculate the approximation error.

### Steps to solve:

#### 1. **Approximate the integral using 4 and 8 rectangles:**
   - **Left Endpoint Rule (LR):**
     \[
     LR = \sum_{i=0}^{n-1} f(x_i) \Delta x
     \]
     where \( \Delta x = \frac{7 - 3}{n} = \frac{4}{n} \), and \( x_i \) are the left endpoints of the subintervals.
   - **Right Endpoint Rule (RR):**
     \[
     RR = \sum_{i=1}^{n} f(x_i) \Delta x
     \]
     where \( x_i \) are the right endpoints.
   - **Midpoint Rule (MR):**
     \[
     MR = \sum_{i=0}^{n-1} f\left( \frac{x_i + x_{i+1}}{2} \right) \Delta x
     \]

#### 2. **Calculate the exact value of the integral:**
   You can evaluate the exact value of the integral using a symbolic integration tool:
   \[
   \int_3^7 x^2 \sqrt{x^3 + 2} \, dx
   \]

#### 3. **Compute the approximation errors:**
   The approximation error for each method is given by:
   \[
   \text{Error} = |\text{Exact Value} - \text{Approximate Value}|
   \]

#### 4. **Discussion:**
   You also need to discuss whether there are functions \( f(x) \) for which the approximation using these methods is exact.

Would you like me to compute the approximations and the exact value for you? 

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Here are some follow-up questions related to the topic:
1. How is the midpoint rule more accurate than the left or right endpoint rule?
2. How does increasing the number of subintervals affect the accuracy of the approximation?
3. What is the trapezoidal rule and how does it compare to the midpoint rule?
4. Can you give an example of a function where the approximation using these methods is exact?
5. How can we measure the efficiency of these approximation methods for complex functions?

**Tip:** When approximating integrals, increasing the number of subintervals will generally lead to a more accurate result.

Approximate the integral ∫₃⁷ x²√(x³ + 2) dx with four approximating rectangles and then eight rectangles, in each case using left endpoints, right endpoints, and midpoints. Then find the exact value of the integral. Compare the approximation of the integral with the exact value of the integral and find the approximation error.

Do you think that, in principle, it is possible to find some functions, f(x), for which the estimation of the integral of f(x) is always equal to the exact value of the integral? Justify your answer in details. Use examples and graphs to support your answer.

Learn how to approximate the integral ∫₃⁷ x²√(x³ + 2) dx using four and eight rectangles with left, right, and midpoint rules. Compare the approximation with the exact value and calculate the error. Understand the principles of numerical integration in this calculus problem, suitable for undergraduate students.

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