题目

Problem

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

$x$	$0.5$	$1.75$	$3$	$4.25$	$5.5$
$y$	$3.479$	$6.101$	$7.448$	$6.823$	$5.182$

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

(a) find an estimate for

\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

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

(2)

解答

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五个 $x$ 值给出四个梯形，步长是 $1.25$ 。端点只算一次，中间三项乘以 $2$ 。

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The strip width is

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

Using the trapezium rule,

\begin{align*} \int_{0.5}^{5.5}f(x)\,\mathrm{d}x &\approx \frac{1.25}{2}\{3.479+5.182\\ &\quad+2(6.101+7.448+6.823)\}\\ =&\,30.878125\\ =&\,30.9 \end{align*}

to one decimal place.

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积分可以拆开： $\int(f(x)+4x)\,\mathrm{d}x=\int f(x)\,\mathrm{d}x+\int4x\,\mathrm{d}x$ 。第一个积分直接用 (a) 的结果。

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Using the result from part (a),

\begin{align*} \int_{0.5}^{5.5}(f(x)+4x)\,\mathrm{d}x &\approx 30.9+\int_{0.5}^{5.5}4x\,\mathrm{d}x\\ =&\,30.9+\left[2x^2\right]_{0.5}^{5.5}\\ =&\,30.9+2(5.5)^2-2(0.5)^2\\ =&\,30.9+60\\ =&\,90.9. \end{align*}