Just an extra note. One thing beyond the scope of what we can learn is multivariate calculus for 3+ spaces. We have so far only focused on 2-D integration. But it extends for more than 2-D. It’s useful for finding volumes and probability theory can use 3 variables to describe an event, as in
\[\mathbb{P}(X \in x, Y \in y, Z \in z) = 0.4\]is more than possible and to find the new density and verify the function’s volume is 1, we use 3-D integration:
\[\int \int \int f(x,y,z)dxdydz\]If you note closely, 2-D integration is really 3-D too!
\[\int \int dxdy = \int_{0}^1 \int \int dxdydz\](sample note.)