Integrating multivariable functions

There are many ways to extend the idea of integration to multiple dimensions: Line integrals, double integrals, triple integrals, surface integrals, etc. Each one lets you add infinitely many infinitely small values, where those values might come from points on a curve, points in an area, points on a surface, etc. These are all very powerful tools, relevant to almost all real-world applications of calculus. In particular, they are an invaluable tool in physics.

With traditional integrals, our "path" was straight and linear (most of the time, we traversed the x-axis). Now we can explore taking integrals over any line or curve (called line integrals).

Rather than integrating along a straight line, such as the x-axis, we will now start thinking about meandering through space. This topic starts with arc length, which leads in nicely to the broader idea of line integration.

You've done some work with line integral with scalar functions and you know something about parameterizing position-vector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.

After introducing line integrals in the context of scalar-valued functions, we see how to integrate along curves which wander through a vector field. This leads to a very beautiful extension of the fundamental theorem of calculus, known as the fundamental theorem of line integrals.

A single definite integral can be used to find the area under a curve. with double integrals, we can start thinking about the volume under a surface!

A single definite integral can be used to find the area under a curve. with double integrals, we can start thinking about the volume under a surface! More generally, double integrals are useful anytime you feel the need to add up infinitely many infinitely small quantities inside some two-dimensional region.

This is about as many integrals we can use before our brains explode. Now we can sum variable quantities in three-dimensions (what is the mass of a 3-D wacky object that has variable density)!

Triple integrals are a way of integreating throughout a three-dimensional region in space.

Just as line integrals give you the ability to add up points on a line, and double integrals give you the ability to add up points in a two-dimensional region, surface integrals are a mechanism for adding points on a curved surface in three-dimensional space.

Flux can be view as the rate at which "stuff" passes through a surface. Imagine a net placed in a river and imagine the water that is flowing directly across the net in a unit of time--this is flux (and it would depend on the orientation of the net, the shape of the net, and the speed and direction of the current). It is an important idea throughout physics and is key for understanding Stokes' theorem and the divergence theorem.

Learn how to compute surface integrals in a vector field, which involves constructing a unit normal vector to a surface. This lets you compute how much fluid flows through a given surface, which provides an intuition much more broadly applicable than just fluids.