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The trapezoidal approximation of the integral \( \int_1^3 e^x \, dx \) using 10 trapezoids gives an approximate value of \( 17.43 \). The graph above shows the trapezoidal approximation (orange areas) under the curve of \( f(x) = e^x \) from \( x = 1 \) to \( x = 3 \), with the trapezoids summing to the approximation.

Do you need further details or have any questions?

Here are five related questions to consider:
1. How does the accuracy of the trapezoidal rule change with increasing \( n \)?
2. What is the exact value of \( \int_1^3 e^x \, dx \), and how close is it to the approximation?
3. Can we use Simpson's Rule to get a more accurate result for this integral?
4. How would the graph look if \( n = 5 \) or \( n = 20 \) trapezoids were used?
5. What are other numerical integration techniques, and how do they compare to the trapezoidal rule?

**Tip**: The accuracy of the trapezoidal rule improves as the number of subintervals increases, but for functions that curve significantly, other methods like Simpson's rule might offer better accuracy.

Provide trapezoidal graph for the integral of e^x from 1 to 3 using n = 10 trapezoids.

Learn how to apply the trapezoidal rule to approximate the integral of e^x from 1 to 3 using 10 trapezoids. Understand the formula, the fundamental theorem of calculus, and explore a graphical representation of this numerical method.

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