# 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 Aug 14, 2020 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. Aug 15, 2020 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}]. Aug 14, 2020 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. Aug 15, 2020 at 10:45
• @pigeon It it the render problem of RegionPlot instead of CubeRoot or Surd Aug 15, 2020 at 12:08
• @pigeon you can also do Area[reg] Aug 15, 2020 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 Aug 15, 2020 at 16:57