Simpson's Rule: Parabolas and the Weights 1, 4, 1

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1. 問1

   A parabola passes through the points $(2, 3)$, $(3, 7)$ and $(4, 5)$. Find the area under it from $x = 2$ to $x = 4$.

   area $=$

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2. 問2

   A river is $18$ m wide. Its depth, measured every $3$ m from one bank to the other, is $0,\ 2,\ 3,\ 3.5,\ 3,\ 1.5,\ 0$ metres. Estimate the cross-sectional area of the river, in square metres, with Simpson's rule.

   area $=$m²

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3. 問3

   A Simpson's-rule estimate with $n = 4$ and $\Delta x = 0.5$ came out as $9$. The heights were $y_0 = 1$, $y_2 = 4$, $y_3 = 6$, $y_4 = 3$, but $y_1$ was lost. Find $y_1$.

   $y_1 =$

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4. 問4

   Approximate $\displaystyle\int_0^1 \frac{dx}{1 + x^2}$ with Simpson's rule using $n = 4$, to four decimal places.

   $S =$

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5. 問5

   Simpson's rule with $n = 2$ is used for $\displaystyle\int_0^2 x^4\,dx$. Find the error, Simpson's estimate minus the exact value, as a fraction.

   error $=$

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