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1. The exact value of the integral \(\int_0^2 \sqrt{1 + x^3} \, dx\), correct to four decimal places, is **3.2413**.

2. Using the trapezium rule with **two strips**, the approximate value of the integral is **3.4142**. The percentage error is given as **5.33%**, which seems to be correct to two decimal places.

3. Using the trapezium rule with **four strips**, the approximate value of the integral is **9.95**. However, this result is marked incorrect. Also, the percentage error is shown as **207.67%**, which is also incorrect.

For the second case (four strips), both the approximate value and percentage error are wrong. Based on the significant overestimation (9.95 compared to the correct value of 3.2413), it seems like there's an issue with the calculation when using four strips.

Would you like help recalculating the values or further details on how to apply the trapezium rule? Here are some questions to guide your exploration:

1. How is the trapezium rule applied for approximating integrals?
2. How do you calculate percentage error for an approximation?
3. How does increasing the number of strips affect the accuracy of the trapezium rule?
4. Why did the four-strip approximation give such a large error?
5. How can you refine the trapezium rule to improve accuracy?

**Tip**: When using numerical methods like the trapezium rule, increasing the number of strips usually improves accuracy, but make sure the setup is correct!

The value of the integral ∫₀² √(1 + x³) dx correct to four decimal places is 3.2413. Using the trapezium rule with two strips gives the approximate value of the integral to be 3.4142, and the percentage error is 5.33%. Using the trapezium rule with four strips gives the approximate value of the integral to be 9.95, but the answer and the percentage error 207.67% are incorrect.

Learn how to approximate the integral ∫₀² √(1 + x³) dx using the trapezium rule with two and four strips. Understand the error calculation and why the four-strip approximation yields a large percentage error. Suitable for advanced high school students.

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