Exercise 8F - 梯形法则应用
1. 基础梯形法则计算
Copy and complete the table below and use the trapezium rule to estimate \(\int_1^3 \frac{1}{x^2 + 1} dx\):
| x | 1 | 1.5 | 2 | 2.5 | 3 |
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| y = \(\frac{1}{x^2 + 1}\) | 0.5 | 0.308 | ? | 0.138 | ? |
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提示：先完成表格，然后使用梯形法则公式计算。
答案：
完成表格：
| x | 1 | 1.5 | 2 | 2.5 | 3 |
|---|
| y = \(\frac{1}{x^2 + 1}\) | 0.5 | 0.308 | 0.2 | 0.138 | 0.1 |
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h = (3-1)/4 = 0.5
Area = ½ × 0.5 × (0.5 + 2(0.308 + 0.2 + 0.138) + 0.1)
= 0.25 × (0.5 + 1.292 + 0.1) = 0.473
2. 平方根函数积分
Use the table below to estimate \(\int_1^{2.5} \sqrt{2x - 1} dx\) with the trapezium rule:
| x | 1 | 1.25 | 1.5 | 1.75 | 2 | 2.25 | 2.5 |
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| y = \(\sqrt{2x - 1}\) | 1 | 1.225 | ? | ? | 1.732 | ? | 2 |
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提示：先完成表格，然后使用梯形法则公式计算。
答案：
完成表格：
| x | 1 | 1.25 | 1.5 | 1.75 | 2 | 2.25 | 2.5 |
|---|
| y = \(\sqrt{2x - 1}\) | 1 | 1.225 | 1.414 | 1.581 | 1.732 | 1.871 | 2 |
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h = (2.5-1)/6 = 0.25
Area = ½ × 0.25 × (1 + 2(1.225 + 1.414 + 1.581 + 1.732 + 1.871) + 2)
= 0.125 × (1 + 15.646 + 2) = 2.331
3. 三次方根函数积分
Copy and complete the table below and use it, together with the trapezium rule, to estimate \(\int_0^2 \sqrt{x^3 + 1} dx\)
| x | 0 | 0.5 | 1 | 1.5 | 2 |
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| y = \(\sqrt{x^3 + 1}\) | 1 | 1.061 | 1.414 | ? | ? |
|---|
提示：先完成表格，然后使用梯形法则公式计算。
答案：
完成表格：
| x | 0 | 0.5 | 1 | 1.5 | 2 |
|---|
| y = \(\sqrt{x^3 + 1}\) | 1 | 1.061 | 1.414 | 1.803 | 3 |
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h = (2-0)/4 = 0.5
Area = ½ × 0.5 × (1 + 2(1.061 + 1.414 + 1.803) + 3)
= 0.25 × (1 + 8.556 + 3) = 3.139
4. 6个梯形的积分计算
Use the trapezium rule with 6 strips to estimate \(\int_1^3 \frac{1}{\sqrt{x^2 + 1}} dx\).
提示：h = (3-1)/6 = 1/3，计算x = 1, 4/3, 5/3, 2, 7/3, 8/3, 3处的y值。
答案：
h = (3-1)/6 = 1/3
| x | 1 | 4/3 | 5/3 | 2 | 7/3 | 8/3 | 3 |
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| y = \(\frac{1}{\sqrt{x^2 + 1}}\) | 0.707 | 0.6 | 0.514 | 0.447 | 0.393 | 0.351 | 0.316 |
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Area = ½ × (1/3) × (0.707 + 2(0.6 + 0.514 + 0.447 + 0.393 + 0.351) + 0.316)
= (1/6) × (0.707 + 4.61 + 0.316) = 0.939
5. 分数函数积分与精度分析
a) Copy and complete the table below and use the trapezium rule to estimate the area bounded by the curve, the x-axis and the lines x = -1 and x = 1.
| x | -1 | -0.6 | -0.2 | 0.2 | 0.6 | 1 |
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| y = \(\frac{1}{x + 2}\) | 1 | 0.714 | ? | ? | 0.385 | ? |
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b) State, with a reason, whether your answer in part a is an overestimate or an underestimate.
提示：先完成表格，然后分析曲线形状判断精度。
答案：
a) 完成表格：
| x | -1 | -0.6 | -0.2 | 0.2 | 0.6 | 1 |
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| y = \(\frac{1}{x + 2}\) | 1 | 0.714 | 0.556 | 0.455 | 0.385 | 0.333 |
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h = (1-(-1))/5 = 0.4
Area = ½ × 0.4 × (1 + 2(0.714 + 0.556 + 0.455 + 0.385) + 0.333)
= 0.2 × (1 + 4.22 + 0.333) = 1.111
b) 这是一个高估值，因为曲线y = 1/(x+2)在区间[-1,1]上是凸的（向上弯曲），梯形完全在曲线下方。