# Result of Integration over ImplicitPlot not as expected

Edit 1:

Issues seem to emerge from within ImplicitRegion as well as Region Plot. Using $$Surd[y^{2},{3}]$$ or avoiding $$y^{2/3}$$ by using $$y^{3}$$ will result in a correct result of Integrate. Area and Integrate seem to return identical results.

In other case both the results of RegionPlot or Integrate[1,Element[{x,y},reg] of the following ImplicitRegion will be wrong, as verified by last result (In case no mistake was done):

$$\text{reg7}=\text{ImplicitRegion}\left[\sin [x]\leq y\leq \cos [x],\left\{\left\{x,0,\frac{3 \pi }{2}\right\},y\right\}\right]$$

The results and double checking them unfold as follows:

$$\text{RegionPlot}[\text{reg7}]$$

$$\text{Integrate}[1,\{x,y\}\in \text{reg7}]$$

$$2 \left(\sqrt{2}-1\right)$$

$$\int_0^{\frac{3 \pi }{2}} \text{Abs}[ \cos [x]-\sin [x]] \, dx$$

$$2 \left(2 \sqrt{2}-1\right)$$

Original Post:

I am trying to Integrate the area mentioned below using ImplicitRegion function, following this resource. This method worked in many other scenario. The plot was accurate, but the integration returned 1/10 which is not true as is verified by the traditional integration. I have tried integration over $$-1\leq y\leq 0$$ and $$0\leq y\leq 1$$ separately and also using Surd[x,3]^2 but things got much weirder. Can any one give a hint how I can improve or correct this?

$$\text{reg}=\text{ImplicitRegion}\left[y\leq x\leq y^{\frac{2}{3}},\{x,\{y,-1,1\}\}\right]$$

$$\text{RegionPlot}[\text{reg}]$$

$$\text{Integrate}[1,y\in \text{reg}]$$

$$\frac{1}{10}$$

$$\int_{-1}^1 \left(y^{\frac{2}{3}}-y\right) \, dy$$

$$\frac{3}{5} \left(1+(-1)^{2/3}\right)$$

• You can format inline code and code blocks by selecting the code and clicking the {} button above the edit window. The edit window help button ? is useful for learning how to format your questions and answers. You may also find this meta Q&A helpful – Michael E2 Aug 14 '20 at 12:47
• The region between Sin[x] and Cos[x] is Min[Sin[x], Cos[x]] <= y <= Max[Sin[x], Cos[x]] instead of Sin[x] <= y <= Cos[x]. I have updated my answer. – cvgmt Aug 15 '20 at 23:10

The region between Sin[x] and Cos[x] is Min[Sin[x], Cos[x]] <= y <= Max[Sin[x], Cos[x]] instead of Sin[x] <= y <= Cos[x]

BTW, since $$\max(f,g)-\min(f,g)=|f-g|$$,that is why your integrate work.

$$\int_a^b |f(x)-g(x)|\,\mathrm{d}x =\int_a^b \left [ \max(f(x),g(x))-\min(f(x),g(x))\right ]\,\mathrm{d}x$$

The set $$\{ (x,y) : \sin(x)\leq y\leq \cos(x) \}$$ is equivalent to $$\{ (x,y) : \sin(x) \leq y\; \text{and} \; y\leq \cos(x) \}$$ which implicitly satisfy a another condition say $$\sin(x)\leq \cos(x)$$

regionOne =
ImplicitRegion[Sin[x] <= y <= Cos[x], {{x, 0, 3/2 \[Pi]}, y}];
regionTwo =
ImplicitRegion[
Min[Sin[x], Cos[x]] <= y <=
Max[Sin[x], Cos[x]], {{x, 0, 3/2 \[Pi]}, y}];
{{RegionPlot[regionOne],
RegionMeasure[regionone]}, {RegionPlot[regionTwo],
RegionMeasure[regionTwo]}}

+++++++++++++++++++++++++++++++++++++++++++++++

reg = ImplicitRegion[y <= x <= CubeRoot[y]^2 && -1 <= y <= 1, {x, y}]
RegionPlot[
DiscretizeRegion[reg, MaxCellMeasure -> {"Length" -> 0.012}]]
Integrate[1, Element[{x, y}, reg]]
Integrate[CubeRoot[y]^2 - y, {y, -1, 1}]

Or

reg = ImplicitRegion[y^3 <= x^3 <= y^2 && -1 <= y <= 1, {x, y}]
Integrate[1, Element[{x, y}, reg]]
• Or Integrate[Surd[y^2, 3] - y, {y, -1, 1}]. – user64494 Aug 14 '20 at 12:21
• @cvgmt RegionPlot would render an incorrect plot in your first option (Same with using Surd[y^2,3] @user64494), even though Integrate result is true, which why I came across it.The second option involves a changing of the formula itself. I am still not able to figure out why I get, either 1/10 instead of 6/5,or a false plot or in other case the formula has to be manipulated. – pigeon Aug 15 '20 at 10:45
• @pigeon It it the render problem of RegionPlot instead of CubeRoot or Surd – cvgmt Aug 15 '20 at 12:08
• @pigeon you can also do Area[reg] – flinty Aug 15 '20 at 13:58
• @flinty Despite seeing this many time, It just never crossed my mind due to the need to figure out how it works. It is just perfect in this scenario! Thanks – pigeon Aug 15 '20 at 16:57