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An artificial lake has the shape illustrated below , with adjacent measurements 20 feet apart. Use suitable numerical method to estimate the surface area of the lake.

enter image description here

I know how to solve this problem we will just assign the adjacent measurements to a parameter $x_i$ and the corresponding lake width to $f(x_i)$ and $x_o=0$ and $f(x_o)=0$ and $x_1=20$ and $f(x_1)=30$ and so on until $f(x_7)$ as the lake is divided into 7 sections, Then we use the trapezoidal as the number of sections is odd.

Now what I don't really understand is why the trapezoidal rule can be applied to this problem and when it isn't viable?

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    $\begingroup$ The trapezoidal rule doesn't require that the number of sections be odd, if that's what you are asking. All you are doing here is approximating the area between each pair of vertical lines by a trapezoid. (I'm not sure if this answers your question or not, as I'm not totally clear where your point of confusion is.) $\endgroup$
    – rogerl
    Commented Jun 1, 2022 at 19:35
  • $\begingroup$ @rogerl I only know 2 rules which are Simpson's rule and the trapezoidal rule and since Simpson's rule requires the number of sections to be even I will use the trapezoidal rule here, My question is when the trapezoidal rule can't be applied ? $\endgroup$
    – Youssef Mohamed
    Commented Jun 1, 2022 at 19:39
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    $\begingroup$ The trapezoidal rule can be applied over any partition of the domain. There are no restrictions as there are with Simpson's rule. $\endgroup$
    – rogerl
    Commented Jun 1, 2022 at 19:44

2 Answers 2

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If you wish to use the trapezoidal rule from integration, you just need to realize that if you slide the lower points so that they are all at height zero, you get the same area (when we approximate the shores linearly).

$$ Area \approx \frac{20}{2}\left(f(x_0) +f(x_7) + 2\sum_{i=1}^6 f(x_i)\right) $$

enter image description here

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  • $\begingroup$ "if you slide the lower points so that they are all at height zero, you get the same area" what do you mean by the same area? $\endgroup$
    – Youssef Mohamed
    Commented Jun 1, 2022 at 20:03
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    $\begingroup$ I mean you get a different shape, but the same area. The area of the trapezoid just depends on the right and left measurements and the 20 ft space between them. $\endgroup$
    – PierreCarre
    Commented Jun 1, 2022 at 20:04
  • $\begingroup$ You can also use Simpson's rule if you leave out the first or last section and compute it separately. $\endgroup$
    – PierreCarre
    Commented Jun 1, 2022 at 20:07
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Let the set $L\subset \Bbb R$ be the lake. Let $L_x=\{y\in\Bbb R:(x,y)\in L\}$. Then from Fubini Theorem $$\mathrm{area}(A) = \int_A dxdy = \int_{-\infty}^\infty \left(\int_{L_x}dy\right)dx = \int_{-\infty}^\infty f(x)dx,$$ where $f$ is defined as in the question, i.e. is the length of $L_x$. Therefore to find the area of $A$ it suffices to calculate the integral of $f$, which can be done numerically.

The calculations explains why PierreCarre are right saying about sliding lower points.

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