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

GROUPS:
 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 }] 
12 days ago
5 Replies
 Michael Rogers 3 Votes 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.
12 days ago
 Thank you Michael that was helpful.
 Vitaliy Kaurov 1 Vote 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
 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}]