The Trapezoidal Rule: The Average of Left and Right Riemann Sums

答え合わせの前に、1問ずつ自分で解いてみましょう。各問に計算用紙と電卓があります。

1. 問1

   To approximate $\displaystyle\int_1^4 \sqrt{x}\,dx$, the interval is cut into $n = 6$ equal steps with points $x_0, x_1, \ldots, x_6$. Find the height $y_4 = f(x_4)$, to three decimal places.

   $y_4 =$

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2. 問2

   Compute the right Riemann sum for $\displaystyle\int_0^2 x^3\,dx$ with $n = 4$ equal steps.

   $R =$

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3. 問3

   Water flows into a reservoir at rate $f(t)$ cubic metres per hour, read every $2$ hours from $t = 0$ to $t = 10$: the readings are $3,\ 5,\ 6,\ 6,\ 4,\ 1$. Estimate $\displaystyle\int_0^{10} f(t)\,dt$, in cubic metres, with the trapezoidal rule.

   $T =$m³

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4. 問4

   For $\displaystyle\int_0^4 \sqrt{x}\,dx$ with $n = 20$ equal steps, find how much larger the right Riemann sum $R$ is than the trapezoidal value $T$, without adding up the terms.

   $R - T =$

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5. 問5

   Approximate $\displaystyle\int_0^1 \frac{4}{1 + x^2}\,dx$ with the trapezoidal rule using $n = 2$ equal steps.

   $T =$

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