title: Benchmark integral vs. Double integral procedure
body: <p>I've got a question about the implementation of a double integral in MATLAB.</p>
  <p>It's known that
  <img src="https://i.stack.imgur.com/xp61K.jpg" alt="enter image description here" /></p>
  <p>Making use of</p>
  <p>k1 = 1E-04:0.001:1E+04;
  k2 = 1E-04:0.001:1E+04;
  k3 = 1E-04:0.001:1E+04;</p>
  <p>the above procedure(calling the formula of F11,F22 and F33) leads to the results shown below:
  <img src="https://i.stack.imgur.com/QBtha.jpg" alt="enter image description here" /></p>
  <p>Now comes the question:</p>
  <p>I would like to achieve the same results by means of a double integral (or nested single integral in k2 and k3) involving only phi11,phi22 and phi33, hence without calling directly the formula for F11, F22 and F33.</p>
  <p>Up to now I'm using this code (restricted to F11 calculation):</p>
  <pre><code>clc,clear all,close all
  k1 = (1E-04:0.01:10000);
  k2 = (1E-04:0.01:10000);
  k3 = (1E-04:0.01:10000);
  F_11_benchmark = ((9/55)*1.453)*(1./(1+k1.^2).^(5/6));
  count = 0;
  F_11 = zeros(1,numel(k1));
  for i = 2:numel(k1)
  count = count + 1;
  phi_11 = @(k2,k3) (1.453./(4.*pi)).*((k2.^2+k3.^2)./((1 + k1(count).^2 + k2.^2+k3.^2).^(17/6)));
  F_11(count) = dblquad(phi_11,k2(i-1),k2(i),k3(i-1),k3(i));
  end
  </code></pre>
  <p>not leading to the same benchmark results shown in the last figure.</p>
  <p>How would you suggest to tackle this problem? I thank you in advance.</p>
  <p>Best regards,
  FPE</p>