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P2 2024 Oct Q1

A Level / Edexcel / P2

IAL 2024 Oct Paper · Question 1

题目

Problem

A continuous curve has equation y=f(x)y=f(x). A table of values of xx and yy for y=f(x)y=f(x) is shown below.

xy0.53.4791.756.10137.4484.256.8235.55.182\begin{array}{c|c} x&y\\\hline 0.5&3.479\\ 1.75&6.101\\ 3&7.448\\ 4.25&6.823\\ 5.5&5.182 \end{array}

Using the trapezium rule with all the values of yy in the given table,

(a) find an estimate for

∫0.55.5f(x) dx\int_{0.5}^{5.5}f(x)\,\mathrm{d}x

giving your answer to one decimal place.

(3)

(b) Using your answer to part (a) and making your method clear, estimate

∫0.55.5(f(x)+4x) dx\int_{0.5}^{5.5}(f(x)+4x)\,\mathrm{d}x
(2)
题目中文翻译

连续曲线的方程为 y=f(x)y=f(x)。下表列出对应的 xx、yy 值：

xy0.53.4791.756.10137.4484.256.8235.55.182\begin{array}{c|c} x&y\\\hline 0.5&3.479\\ 1.75&6.101\\ 3&7.448\\ 4.25&6.823\\ 5.5&5.182 \end{array}

使用表中全部 yy 值和梯形法则：

(a) 求以下积分的估计值，答案精确到1位小数。(3)

∫0.55.5f(x) dx\int_{0.5}^{5.5}f(x)\,\mathrm{d}x

(b) 使用 (a) 的答案并清楚展示方法，估计：(2)

∫0.55.5(f(x)+4x) dx\int_{0.5}^{5.5}\big(f(x)+4x\big)\,\mathrm{d}x

解答

(a)

解法一

思路

展开

五个 xx 值给出四个梯形，步长是 1.251.25。端点只算一次，中间三项乘以 22。

答题过程

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The strip width is

h= 5.5−0.54=1.25.\begin{align*} h=&\,\frac{5.5-0.5}{4}=1.25. \end{align*}

The interior ordinates sum to 6.101+7.448+6.823=20.3726.101+7.448+6.823=20.372. Using the trapezium rule,

 ∫0.55.5f(x) dx≈ 0.625[3.479+5.182 +2(20.372)]= 30.878125≈ 30.9\begin{align*} &\,\int_{0.5}^{5.5}f(x)\,\mathrm{d}x\\ \approx&\,0.625\big[3.479+5.182\\ &\,\hspace{2pt}+2(20.372)\big]\\ =&\,30.878125\\ \approx&\,30.9 \end{align*}

to one decimal place.

(b)

解法一

思路

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利用积分的线性性质把两个被积项分开；第一项直接用 (a) 的近似结果，第二项积分 4x4x。

答题过程

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Using the result from part (a),

 ∫0.55.5(f(x)+4x) dx≈ 30.9+∫0.55.54x dx= 30.9+[2x2]0.55.5= 30.9+2(5.5)2−2(0.5)2= 30.9+60= 90.9.\begin{align*} &\,\int_{0.5}^{5.5}\big(f(x)+4x\big)\,\mathrm{d}x\\ \approx&\, 30.9+\int_{0.5}^{5.5}4x\,\mathrm{d}x\\ =&\,30.9+\big[2x^2\big]_{0.5}^{5.5}\\ =&\,30.9+2(5.5)^2-2(0.5)^2\\ =&\,30.9+60\\ =&\,90.9. \end{align*}

解法二

思路

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积分 4x4x 等于直线 y=4xy=4x 下的梯形面积，因此不用另作积分；仍须承接 (a) 的估计值。

答题过程

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On 0.5⩽x⩽5.50.5\leqslant x\leqslant5.5, the endpoint heights of the line y=4xy=4x are 22 and 2222. Its area is exactly

T= 12(2+22)(5.5−0.5)= 60.\begin{align*} T=&\,\frac12(2+22)(5.5-0.5)\\ =&\,60. \end{align*}

Using part (a),

 ∫0.55.5(f(x)+4x) dx≈ 30.9+T= 90.9.\begin{align*} &\,\int_{0.5}^{5.5}\big(f(x)+4x\big)\,\mathrm{d}x\\ \approx&\,30.9+T\\ =&\,90.9. \end{align*}

也可以把这个梯形分成面积 2×5=102\times5=10 的矩形和面积 12×5×20=50\frac12\times5\times20=50 的三角形，仍得到 6060。