I am trying to find an equation that estimates the following integral:
$$\int_0^1 \int_0^1 e^{(x+y)^2} \,dx\,dy$$
where I am given a list of different uniform RV's $m_1,m_2,\ldots$ and $n_1,n_2,\ldots$.
I'm not quite sure how to go about evaluating this double integral?
I know the integral for a single integral over that interval is a summation over n, but how does it work for two integrals where the function is already given?
Any insight would be great!
thank you
On
i want to give you the answer only, which I found from a website. I am giving you the link. Hope it will help you. Actually, I don't know how to solve this problem. https://www.wolframalpha.com/input/?i=%E2%88%AB%280%2C1%29%E2%88%AB%280%2C1%29+e%5E%28%28x%2By%29%5E2%29+dxdy
The measure $dx\,dy$ is is the uniform distribution on $[0,1]\times[0,1].$ So (assuming the uniform random variables are independent—and you should have said something about their joint distribution) the sample mean $$ \frac 1 k \sum_{i=1}^k e^{(m_i+n_i)^2} $$ should approximate the value of the integral.