Thinking about integrals
*This is only my way of thinking about integrals and not the only way.
Taking a single integral from the function f(x) gives us the area or "mass" between the x-axis and the f(x) graph. You can think of this as drawing many single lines from graph f(x) on to the x-axis and seeing how long the lines are for each point. Then summing together these different length lines you get the area or "mass" of the graph between specific x values.
Taking a double integral over the two variable function f(x,y) gives us the volume or "mass" between a plane on x- and y-axis and the function f(x,y). You can think of this as drawing many small rectangles as you can on to the plane(for example to the area your house takes when looked at above) and seeing how high each of these squares can grow before hitting the graph f(x,y) (hitting the height of your roof at that point). If we color these rectangles to the plane(or a pic of your house from above) by how high they were able to get, we get map that tells the "density of the volume" on each point. Then summing these different height(/color) rectangles together we get the volume or "mass" of the function over the area defined by x and y.
Taking a triple integral over three variable function f(x,y,z) gives us the density or "mass" between a space defined by x, y and z in 3D space and it's function f(x,y,z). You can think of this as counting the air mass inside your home. The x, y and z define the inside space in your home and when we split your home into many small cubes each of those cubes have their own position marked with x, y and z coordinates and function f(x,y,z) gives the number of air molecules inside that cube. Of course your house won't have the same amount of air molecules inside every cube so we get a 3D "density distribution" of the whole space. Then summing all these cubes together we get the density or "mass" inside the whole space defined by x, y and z.
Again this is just a single example and these depend highly on the thing we are calculating so for example if we have an area and the function f(x,y) gives the density in the point x,y then obviously "volume" is not the term that should be used:). Also integrals can give negative values when again it's harder to visualize. But I found thinking about these in terms of "mass" was more intuitive.