# Result of NIntegrate ploting to list

I’ve been working on a project where I need to make two Plots. Those plots are of the form:

Plot[
NIntegrate[f[x],{x,0,r}],
{r,0,1}
]
Plot[
NIntegrate[-f[x],{x,0,r}],
{r,0,1}
]

Indeed one of the plots is just the minus the other and I’m wasting computational time doing both the numerical integrals. I then thought that I could create a list of the values of the first plot and then plot that list. Here’s what I did:

s = {};
Plot[
Last[Last[
s = Append[s, {r, NIntegrate[f[x], {x, 0, r}]}]
]],
{r, 0, 2}]
ListLinePlot[s/. {x_, y_} -> {x, -y}]

This however doesn’t work the way I wanted. The problem is that aside the values given by NIntegrate[f[x],{…}] there are other values in the list s. Let me present a working example. Consider that the the function that I’m integrating is f[x]=x. Substituting in my code I get the following values for the list s:

{{0.0000408571, 8.34653*10^-10}, {4.08163*10^-8,
8.32986*10^-16}, {0.0392573, 0.000770569}, {0.0818167,
0.00334699}, {0.121556, 0.0073879}, {0.160515,
0.0128826}, {0.202777, 0.0205592}, {0.242218, 0.0293348}, {0.284962,
0.0406016}, {0.326926, 0.0534402}, {0.366069,
0.0670033}, {0.408515, 0.0834422}, {0.44814, 0.100415}, {0.486986,
0.118578}, {0.529134, 0.139991}, {0.568461, 0.161574}, {0.611091,
0.186716}, {0.652941, 0.213166}, {0.691971, 0.239412}, {0.734303,
0.2696}, {0.773815, 0.299394}, {0.816628, 0.333441}, {0.858662,
0.368651}, {0.897876, 0.403091}, {0.940392, 0.442169}, {0.980088,
0.480286}, {1.019, 0.519184}, {1.06122, 0.563096}, {1.10062,
0.605682}, {1.14332, 0.65359}, {1.18524, 0.702397}, {1.22434,
0.749505}, {1.26674, 0.802319}, {1.30632, 0.853242}, {1.34513,
0.904683}, {1.38723, 0.962206}, {1.42652, 1.01747}, {1.4691,
1.07913}, {1.50887, 1.13834}, {1.54785, 1.19793}, {1.59014,
1.26428}, {1.62961, 1.32782}, {1.67238, 1.39843}, {1.71437,
1.46954}, {1.75354, 1.53745}, {1.79601, 1.61283}, {1.83567,
1.68484}, {1.87454, 1.75695}, {1.91671, 1.8369}, {1.95607,
1.9131}, {2., 2.}, {0.0196287, 0.000192643}, {0.060537,
0.00183237}, {0.101686, 0.00517005}, {0.141036,
0.00994551}, {0.181646, 0.0164977}, {0.222498, 0.0247526}, {1.97803,
1.95631}, {0.00981436, 0.0000481608}, {0.0498972,
0.00124486}, {0.0917515, 0.00420917}, {0.131296,
0.00861927}, {0.171081, 0.0146343}, {0.212637, 0.0226073}, {1.96705,
1.93465}, {0.029443, 0.000433445}, {0.0711769,
0.00253307}, {0.111621, 0.00622962}, {0.150775,
0.0113666}, {1.98902, 1.97809}, {0.0049072,
0.0000120403}, {0.0445772, 0.000993565}, {0.0867841,
0.00376574}, {0.126426, 0.00799173}, {0.165798,
0.0137445}, {0.207707, 0.0215711}, {1.96156, 1.92386}, {0.0245358,
0.000301004}, {0.0658569, 0.00216857}, {0.106654,
0.0056875}, {0.145905, 0.0106442}, {1.98353, 1.96719}, {0.0147215,
0.000108362}, {0.0552171, 0.00152446}, {0.0343502,
0.000589967}, {1.99451, 1.98903}, {0.00245362,
3.01013*10^-6}, {0.0419173, 0.000878529}, {0.0843004,
0.00355328}, {0.123991, 0.00768685}, {0.163157, 0.01331}, {0.205242,
0.0210621}, {1.95881, 1.91848}, {0.0220823,
0.000243813}, {0.063197, 0.00199693}, {0.10417,
0.00542569}, {0.143471, 0.0102919}, {1.98078, 1.96174}, {0.0122679,
0.0000752512}, {0.0525571, 0.00138113}, {0.0318966,
0.000508696}, {1.99176, 1.98356}, {0.00736078,
0.0000270905}, {0.0269894, 0.000364214}, {0.0171751,
0.000147492}, {1.99725, 1.99451}, {0.00122683,
7.52557*10^-7}, {0.0405873, 0.000823665}, {0.0830586,
0.00344936}, {0.122773, 0.00753664}, {0.161836,
0.0130954}, {0.204009, 0.0208099}, {1.95744, 1.91579}, {0.0208555,
0.000217475}, {0.061867, 0.00191376}, {0.102928,
0.0052971}, {0.142253, 0.010118}, {1.97941, 1.95903}, {0.0110412,
0.0000609535}, {0.0512272, 0.00131211}, {0.0306698,
0.000470318}, {1.99039, 1.98083}, {0.00613399,
0.0000188129}, {0.0257626, 0.000331857}, {0.0159483,
0.000127174}, {1.99588, 1.99177}, {0.00368041,
6.77271*10^-6}, {0.0134947, 0.0000910539}, {0.00858757,
0.0000368732}, {1.99863, 1.99726}, {0.000613436,
1.88152*10^-7}, {0.0399223, 0.000796896}, {0.0824376,
0.00339798}, {0.122165, 0.00746208}, {0.161176,
0.0129888}, {0.203393, 0.0206844}, {1.95676, 1.91445}, {0.0202421,
0.000204871}, {0.061202, 0.00187284}, {0.102307,
0.00523338}, {0.141644, 0.0100316}, {1.97872, 1.95767}, {0.0104278,
0.0000543691}, {0.0505622, 0.00127827}, {0.0300564,
0.000451694}, {1.9897, 1.97946}, {0.0055206,
0.0000152385}, {0.0251492, 0.000316242}, {0.0153349,
0.00011758}, {1.99519, 1.9904}, {0.00306702,
4.70329*10^-6}, {0.0128813, 0.0000829644}, {0.00797418,
0.0000317937}, {1.99794, 1.99588}, {0.00184023,
1.69322*10^-6}, {0.00674739, 0.0000227636}, {0.00429381,
9.21838*10^-6}, {1.99931, 1.99863}}

So after the first digit of the list of lists reaches 22 – the plot ranges from r=0r=0 to r=2r=2 – the list has other values… What am I doing wrong?

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2

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Here is another approach based on excellent Coolwater’s idea:

f[x_] = x;
s = NDSolve[{g[0] == 0, g'[x] == f[x]}, g, {x, 0, 1}][[1, 1, 2]];
ListLinePlot[Transpose[{Flatten@#[“Grid”], #[“ValuesOnGrid”]}] &[s], Mesh -> All]

(I directly extract all the interpolation points from the InterpolatingFunction, see this answer for additional information.)

As to the Plot-related part of the question, see this answer of mine for detailed explanation how Plot works. In short, what had puzzled you is adaptive refinement which can be turned off by setting MaxRecursion->0 option for Plot. For getting correct plot with your method you should simply Sort the list of points:

ListLinePlot[Sort[s /. {x_Real, y_Real} :> {x, -y}], Mesh -> All]

But there is simpler method to achieve what you want:

f[x_] = x;
pl1 = Plot[NIntegrate[f[x], {x, 0, r}], {r, 0, 2}, PlotRange -> All];
pl1 /. l_Line :> {{Blue, l}, {Red, l /. {x_Real, y_Real} :> {x, -y}}}

Very nice approach to understand what’s going on behind the scenes. Thank you. I still wonder, though, why the extra terms on the list from my example.
– PML
Feb 3 ’14 at 21:26

@PML I updated the unswer.
– Alexey Popkov
Feb 4 ’14 at 7:55

– PML
Feb 4 ’14 at 15:33

This should work

{Plot[{#, -#}, {r, 0, 1}]} &[NDSolve[{g[0] == 0,
g'[x] == f[x]}, g, {x, 0, 1}][[1, 1, 2]][r]]

I believe that indeed works. But could you help me understand your solution, specifically, the part …[[1,1,2][r]]?
– PML
Feb 3 ’14 at 19:13

2

@PML NDSolve returns the solution with structure: {{g->InterpolatingFunction[{{0.,1.}},<>]}}. Part [[1, 1, 2]] just extracts the InterpolatingFunction and [r] passes the variable r as an argument to it. Then Plot[<...>,{r,0,1}] can make a plot of the InterpolatingFunction as function of r.
– Alexey Popkov
Feb 3 ’14 at 20:28