f(x) =3x a la2 +1  intervalos 0-3 rectangulos 8

The function you're referring to is \( f(x) = 3x^2 + 1 \), and you want to estimate the area under this curve from \( x = 0 \) to \( x = 3 \) using 8 rectangles. This method is likely the **Riemann sum** method, which approximates the integral of a function over an interval.

### Steps to solve:
1. **Interval of Integration**: From \( x = 0 \) to \( x = 3 \).
2. **Number of Rectangles (n)**: 8 rectangles.
3. **Width of each rectangle (\(\Delta x\))**:
   \[
   \Delta x = \frac{{b - a}}{n} = \frac{3 - 0}{8} = 0.375
   \]
4. **Left or Right Endpoint**: You can use either left-endpoints, right-endpoints, or midpoints to evaluate the height of the rectangles. Let's proceed with **right endpoints**.

For each \( i \)-th rectangle, the right-endpoint of \( x_i \) will be:
\[
x_i = a + i\Delta x = 0 + i \times 0.375 = 0.375i
\]
where \( i \) is from 1 to 8.

The height of each rectangle will be \( f(x_i) = 3x_i^2 + 1 \). So, the Riemann sum is:
\[
\text{Riemann Sum} = \sum_{i=1}^{8} f(x_i) \Delta x
\]
Now let's compute that step-by-step:

- \( x_1 = 0.375 \), \( x_2 = 0.75 \), \( x_3 = 1.125 \), ..., \( x_8 = 3 \).
- Then, find the height \( f(x_i) = 3x_i^2 + 1 \) for each \( i \).
- Finally, multiply by the width (\(\Delta x = 0.375\)) and sum them up.

Let me calculate that for you.The Riemann sum approximation for the area under the curve \( f(x) = 3x^2 + 1 \) from \( x = 0 \) to \( x = 3 \), using 8 rectangles and the right-endpoint method, is approximately **35.27**.

Would you like more details about this process or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the result change if we used the left-endpoint method?
2. How does increasing the number of rectangles affect the approximation?
3. What is the exact integral value of \( f(x) = 3x^2 + 1 \) from 0 to 3?
4. Can we use the midpoint method to improve the approximation?
5. How do Riemann sums relate to definite integrals?

**Tip**: Using more rectangles in a Riemann sum leads to a more accurate approximation of the integral.

f(x) =3x a la2 +1 intervalos 0-3 rectangulos 8

Explore how to approximate the area under the curve of the quadratic function f(x) = 3x^2 + 1 over the interval [0, 3] using 8 rectangles. This problem demonstrates the Riemann sum method, with calculations using right endpoints, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation