Small error in "One Dimensional MC Integration"

[domitry]

raytracing.github.io/books/RayTracingTheRestOfYourLife.html

Line 200 in 2460c2e

$$ I = \int_{0}^{2} x^2 dx $$

While the interval of integration is [0, 2], corresponding sample code is as below:

raytracing.github.io/books/RayTracingTheRestOfYourLife.html

Lines 226 to 233 in 2460c2e

	int N = 1000000;
	double sum;
	for (int i = 0; i < N; i++) {
	auto x = random_double(0,2);
	sum += x*x;
	}
	std::cout << std::fixed << std::setprecision(12);
	std::cout << "I = " << sum/N << '\n';

Should sum/N be replaced to 2*sum/N?

[petershirley]

Let's try f(x) = 1 as the integrand. That gives "I = 2" as the analytic answer. Then the sum += 1 should be what we have. So sum = N. Then we divide by N and get I=1. OK yes something is wrong here. We need to divide by p(x) which is 1/2, so multiply by 2. I agree with the fix.