How Do Integrals Behave In Higher Dimensions? As I started looking into Linear Algebra and reading Flatlands: A Romance of Many Dimensions after learning Calculus, I wondered how integrals act in higher dimensions. So here I am on Stack Exchange. How do integrals behave in higher dimensions?
So far you've been taking integrals with respect to one variable, whether that be $x$ or $y$. An integral's main purpose is to provide the area under the curve. 

Of course in higher dimensions, for example, 3$D$, that area is known as "volume".

Now you actually have 2 axis, the $x$ and $y$ axis, and so you will encounter what's known as a double integral, where you integrate with respect to both axes.

Right now you are fimilar with single integrals, or integrals pertaining to only one axes.

Here is a visual representation. 



Here you can integrate via $x$ or $y$, and you will get the area of the blue region.

In 3D, you will have two set of axes, and your output will be on the third axes (usually $z$)

Notice how here the axes are $x$ and $y$ and your output is in the $z$ axis. Here you have volumn between two sets of points.



Anyways, that's the basic idea. Integrals.
How bout this? Your velocity as a function of time is $&lt;2t,t^2&gt;$. Then your location is $\int &lt;2t,t^2&gt;\, dx=&lt;t^2,t^3/3&gt;+C$.