Area under parametric curves

Question

I would like to find the area under the following parametric plot in Wolfram Mathematica:

F[j_] := NIntegrate[2 Sqrt[1 - j^2/r^2], {r, j, 1}]

ParametricPlot[{{F[j], F[j] + 2 Pi*j}, {F[j] + 2 Pi *j, F[j]}}, {j, 0, 
  1}]

I tried to do it via the calculus formula $A=\int_0^1 y(j)x'(j)dj$, using NIntegrate, but Mathematica returns some limit errors.

I would really appreciate if somebody had an easy way to do this.

Answer 1

You have two identical areas, just rotated by 90 degrees, plus a 2*2 block in the lower left corner. Create a Region from a Polygon to approximate one of those areas, multiply that area by 2, add 2*2 and you have the area under your parametric curve.

points=Join[{{0,2}},Table[{F[j], F[j] + 2 Pi*j}, {j, 0, 1,1/10}]]
area=2*Area[Region[Polygon[points]]]+2*2

which returns

{{0,2},{2.,2.},{1.69585,2.32417},{1.41182,2.66845},{1.14822,3.03317},
 {0.905607,3.41888},{0.684853,3.82645},{0.487246,4.25716},
 {0.314727,4.71296},{0.170398,5.19695},{0.0599315,5.7148},{0.,6.28319}}

and

9.9006

Change that 1/10 step size to 1/100 or 1/200 and you should have a good enough approximation of your area because your parametric curve is nice and smooth.

Test this brutally before you even think of trusting this.