Explain why they both give the same answer, not which one to choose. Here\'s yet

Question

Explain why they both give the same answer, not which one to choose.

Here's yet another polar curve: Clear [x, y, r, t]; r[t_] = 5 + 3 Cos [3 t] -1.5 Sin[5t]; x[t_] = r[t] Cos[t]; y[t_] = r[t] Sin[t]; tlow = 0; thing = 2 pi original = ParametricPlot[{x[t], y[t]}, {t, tlow, thigh}, PlotStyle rightarrow {{Blue, Thickness [0.02] }}, AspectRatio rightarrow Automatic, Axes Label rightarrow {"x", "y"}] As t advances from 0 to 2 pi, this curve plots out in the counterclockwise manner with no segment repeated. Explain why you can calculate the area inside this curve by calculating y[t] x'[t] dt": Out[1472]=96.2113 Or by calculating r[t]2/2 dt: Out[1473]= 96.2113

Explanation / Answer

The first one is formula for finding area when parametric coordinates of a curve are given while second one is formula for calculating area when polar coordinates are given. since in the question the x and y coordinates are given in form of r(t), the could be considered as parametric coordinates thus both give same results. For more detail, refer any textbook for derivation of formula for area of a curve with polar coordinates and parametric coordinates. please rate as lifesaver

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