Result of NIntegrate ploting to list

Question

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 $2$ - the plot ranges from $r=0$ to $r=2$ - the list has other values... What am I doing wrong?

Answer 1

This should work

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

Answer 2

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}}}