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[✓] Area under the curve of a Parametric Plot function?

Posted 11 months ago
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I am looking to find the area under the curve of the a parametric plot function I tried to use a simple example but it's giving me two answers for each function rather than one when I use NIntegrate. Is there another way to get the area under the curve of the parametric plot ? Thank you

ParametricPlot[{Sin[t], Sin[2 t]}, {t, 0, 1.57 }]
NIntegrate[{Sin[t], Sin[2 t]}, {t, 0, 1.57 }]
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If the curve $C$ is above the x axis and moves from left to right, then $A = \int_C y \; dx$:

ParametricPlot[{Sin[t], Sin[2 t]}, {t, 0, Pi/2}]
With[{x = Sin[t], y = Sin[2 t]}, Integrate[y D[x, t], {t, 0, Pi/2}]]

1.57 and NIntegrate may replace Pi/2 and Integrate if needed.

Posted 11 months ago

Thank you Michael that was helpful.

Not as instructive calculus but still as verification path ParametricRegion:

Area[ParametricRegion[{r Sin[t], r Sin[2 t]}, {{t, 0, Pi/2}, {r, 0, 1}}]]

2/3

Posted 11 months ago

Thank you Vitaliy that's another helpful approach.

Another way, less general than previous ones... Notice first that

Sin[2 t] // TrigExpand

gives 2 Cos[t] Sin[t] Then perform a revealing change of variable:

x[t_] := Sin[t];
y[t_] := 2 x[t] x'[t];
y[ArcSin[x]]
Integrate[y[ArcSin[x]], {x, 0, 1}]
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