https://file.aiquickdraw.com/mathbot/file/1728529425196tao4yu5n.jpg

To solve this problem using the **Trapezoidal Rule**, we will approximate the definite integral \( \int_0^8 f(x) \, dx \) based on the values provided in the graph of \( f(x) \).

### Trapezoidal Rule formula:
The Trapezoidal Rule approximates the integral by dividing the interval into smaller subintervals and averaging the areas of the trapezoids formed. For a general function \( f(x) \) over \( n \) intervals, the approximation is given by:

\[
T_n = \frac{b - a}{2n} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n) \right)
\]

Where:
- \( a \) and \( b \) are the limits of integration (in this case, \( a = 0 \) and \( b = 8 \)),
- \( f(x_0), f(x_1), \dots, f(x_n) \) are the function values at the endpoints and the points in between.

Here, the graph shows 9 points corresponding to \( x = 0, 1, 2, 3, 4, 5, 6, 7, 8 \). The values of \( f(x) \) at these points can be read from the graph:

- \( f(0) = 0 \)
- \( f(1) = 1 \)
- \( f(2) = 3 \)
- \( f(3) = 6 \)
- \( f(4) = 7 \)
- \( f(5) = 7 \)
- \( f(6) = 6 \)
- \( f(7) = 5 \)
- \( f(8) = 4 \)

Now, applying the **Trapezoidal Rule**:

\[
T_4 = \frac{8 - 0}{2 \times 4} \left( f(0) + 2f(1) + 2f(2) + 2f(3) + 2f(4) + 2f(5) + 2f(6) + 2f(7) + f(8) \right)
\]

\[
T_4 = \frac{8}{8} \left( 0 + 2(1) + 2(3) + 2(6) + 2(7) + 2(7) + 2(6) + 2(5) + 4 \right)
\]

\[
T_4 = 1 \left( 0 + 2 + 6 + 12 + 14 + 14 + 12 + 10 + 4 \right)
\]

\[
T_4 = 74
\]

Therefore, the estimate of the integral using the Trapezoidal Rule is:

\[
T(4) = 74
\]

Would you like any further details or have questions? Here are 5 related questions to deepen your understanding:

1. How does the Trapezoidal Rule differ from Simpson's Rule for approximating integrals?
2. Why does the Trapezoidal Rule use the function values at the boundaries of each subinterval?
3. How would increasing the number of subintervals affect the accuracy of the Trapezoidal Rule?
4. Could we estimate this integral using the Midpoint Rule, and how would that change the result?
5. What are the possible sources of error when using the Trapezoidal Rule for approximating integrals?

**Tip**: The more intervals you use, the more accurate the Trapezoidal Rule becomes, especially for functions with curvature between intervals.

Compute the following estimate of ∫ f(x) dx using the graph in the figure. T(4)

Learn how to estimate the integral of f(x) from 0 to 8 using the Trapezoidal Rule, applied to T(4). This problem uses values from a graph to compute the integral, perfect for students in Grades 11-12 studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation