This 16-minute video shows that the line integral of a scalar field is independent of path direction.

This 10-minute video lesson provides a second example of using path independence of a conservative vector field to solve a line integral.

In this tutorial, we will learn to approximate differentiable functions with polynomials. Beyond just being super cool, this can be useful for approximating functions so that they are easier to calculate, differentiate or integrate. So whether you will have to write simulations or become a bond trader (bond traders use polynomial approximation to estimate changes in bond prices given interest rate changes and vice versa), this tutorial could be fun. If that isn't motivation enough, we also come up with one of the most epic and powerful conclusions in all of mathematics in this tutorial: Euler's identity.

This 10-minute video lesson provides an introduction to the arithmetic and geometric series.

This 10-minute video lesson looks at finding the sum of an infinite geometric series.

This 10-minute video lesson take the revolution around something other than one of the axes.

This 4-minute video lesson looks at the last part of the problem in part 7 (see previous video lesson).

This video looks at the example showing how to find the volume of a solid of revolution (constructed by rotating around the x-axis) using the shell method (this could have been done with the disk method as well).

You want to rotate a function around a vertical line, but do all your integrating in terms of x and f(x), then the shell method is your new friend. It is similarly fantastic when you want to rotate around a horizontal line but integrate in terms of y.

This 9-minute video lesson uses the \shell method\ to rotate about the y-axis.

This video covers using the shell method to rotate around a vertical line.

In this tutorial, we'll think about how we can find the area under a curve. We'll first approximate this with rectangles (and trapezoids)--generally called Riemann sums. We'll then think about find the exact area by having the number of rectangles approach infinity (they'll have infinitesimal widths) which we'll use the definite integral to denote.

This 16-minute video lesson covers the Calculus-Derivative: Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line).

This series of videos focusing on calculus covers using definite integrals with the shell and disc methods to find volumes of solids of revolution.

This 8-minute video lesson presents an intuition (but not a proof) of the Squeeze Theorem. [Calculus playlist: Lesson 7 of 156]

If a function is always smaller than one function and always greater than another (i.e. it is always between them), then if the upper and lower function converge to a limit at a point, then so does the one in between. Not only is this useful for proving certain tricky limits (we use it to prove lim (x Œ_ 0) of (sin x)/x, but it is a useful metaphor to use in life (seriously). :) This tutorial is useful but optional. It is covered in most calculus courses, but it is not necessary to progress on to the "Introduction to derivatives" tutorial.