How to interpret a double integral

Question

If you have an integral $\int_c^{d}\int_{a}^{b} f(x_1,x_2)\,dx_2\, dx_1$. I am not sure how to visualize this.I know that you are adding two dimensional rectangles but I cannot see the relationship between the formula and the visualization. Do you basically add all the rectangles in the x2 direction first and get a function of x1 and then add all the rectangles in the x2 direction to get a function of x2? It's easy to interpret a single variable integral but I am not sure what's actually being done in a double integral.I know it's the volume under a particular function in the xyz plane but I cannot determine the "algorithm" that is performed to actually compute that volume.

Answer 1

You can think $f(x_1, x_2)$ as height function over $X_1- X_2$ plane. $$\int_c^d\int_a^b f(x_1,x_2) dx_2dx_1$$ will be volume of solid bounded by
1- graph $f$ and 2- $X_1-X_2$ plane 3- $X_2=a \text{ to }X_2= b$ and 4- $X_1=c \text{ to } X_1= d$. This is just a vague visualization. Because $f$ may not be graph globally....and it may be negative... But then you can make partition of domain suitably and break the integration... and Each partition, you can think given integration as SIGNED volume...