Integral over implicit function Suppose the equation is given:
$$y^5+2y^4-7y^3+y-x=0$$
(or any other equation that cannot be expressed explicitly).
Let the solutions be implicitly given as $y=g(x)$. Is there an approach to solve the integral
$$\int\limits_0^1 x g(x) \, dx$$
either numerically or analytically?
 A: Your equation isn't as random as you might think, but rather of the form $x(y)=P_n(y)$, where $P_n$ 
is a polynomial of the n-th degree. Now, $\displaystyle\int x\cdot y~dx=\int x(y)\cdot y~dx(y)=\int P_n(y)\cdot y\cdot P_n'(y)~dy$ 
where the new limits are the values of y for which $x=P_n(y)$ takes the values $0$ and $1$ respectively 
with the added mention that x must be bijective on that interval. In this particular case, there are 
three such choices. Unfortunately, only $P_n(y)=0$ can be expressed in terms of radicals. The other 
is an unsolvable quintic. Either way, we are dealing with $F\big(y_1\big)-F\big(y_0\big)$, where F is a polynomial 
of degree $2n+1$.
A: Hard to get a general recipe. Numerically, we see from the graph of $x$ that the range $[0,1]$ implies we are interested in the "branch" that goes from $y_0\approx -0.36$ to $y_1\approx -0.58$ ($y=g(x)$ is actually multivalued). When that piece of the function $y$ is multiplied by $x$ we get something like the red line in the graph. 

Its integral (orange area) can be obtained numerically (Montecarlo), eg, in Java :
  public static double integral(int times) {
    Random rand = new Random();
    int count = 0;
    for (int t = 0; t < times; t++) {
      double x = rand.nextDouble();
      double y = -rand.nextDouble();
      y /= x;
      if (y < -1.0)
        continue;
      double x2 = Math.pow(y, 5) + 2 * Math.pow(y, 4) - 7 * Math.pow(y, 3) + y;
      if (x2 < x)
        count++;
    }
    return - count / (double) times;

  }

I get $I\approx 0.266$. This is not very tested, and certainly it's not optimized.
