https://file.aiquickdraw.com/mathbot/file/17314042032771nogn362.jpg

Give me the Reinmann sum of this function: 

To calculate the left endpoint Riemann sum for the function \( y = \frac{x^2}{9} \) over the interval \([3, 7]\), we’ll follow these steps:

1. **Identify the interval and subintervals**:
   - Interval: \([3, 7]\)
   - Subintervals: Based on the graph, there are five rectangles.

2. **Determine the width of each rectangle (\(\Delta x\))**:
   - The interval length is \(7 - 3 = 4\).
   - With 5 rectangles, \(\Delta x = \frac{4}{5} = 0.8\).

3. **Calculate the left endpoint values**:
   - For each rectangle, we use the left endpoint to calculate the height. The left endpoints will be:
     - \(x_1 = 3\)
     - \(x_2 = 3.8\)
     - \(x_3 = 4.6\)
     - \(x_4 = 5.4\)
     - \(x_5 = 6.2\)

4. **Evaluate the function at each left endpoint**:
   - \( f(x) = \frac{x^2}{9} \)
   - \( f(3) = \frac{3^2}{9} = 1 \)
   - \( f(3.8) = \frac{3.8^2}{9} \approx 1.60 \)
   - \( f(4.6) = \frac{4.6^2}{9} \approx 2.35 \)
   - \( f(5.4) = \frac{5.4^2}{9} \approx 3.24 \)
   - \( f(6.2) = \frac{6.2^2}{9} \approx 4.27 \)

5. **Calculate the Riemann sum**:
   - Riemann Sum \( = \sum f(x_i) \cdot \Delta x \)
   - \( = (1 + 1.60 + 2.35 + 3.24 + 4.27) \times 0.8 \)
   - \( = 12.46 \times 0.8 \approx 9.97 \)

The left endpoint Riemann sum for \( y = \frac{x^2}{9} \) on \([3, 7]\) is approximately **9.97**.

Would you like a detailed breakdown of any specific steps or have further questions?

Here are some related questions for further exploration:
1. How does the result change with a right endpoint Riemann sum?
2. How would a midpoint Riemann sum affect the result?
3. What happens if we increase the number of rectangles (subintervals)?
4. Can we calculate the exact integral of \( y = \frac{x^2}{9} \) over \([3, 7]\) and compare it?
5. How does the Riemann sum relate to the area under the curve?

**Tip:** Increasing the number of rectangles in a Riemann sum generally improves the approximation of the integral.

Give me the Riemann sum of this function:

Learn how to calculate the left endpoint Riemann sum for the function y = x^2/9 over the interval [3, 7] with five rectangles. This solution covers essential calculus concepts including Riemann sums, function evaluation at left endpoints, and calculating Δx. Suitable for high school and introductory college calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation