The remaining lectures in this course give us the framework for applying integration to the real world. Everything that we do here shows up time and again in engineering - Integrals as Net Change allow us to find the total of a changing value; Volumes of Rotation let us find the capacity of conduits and volumes of columns; Arclength lets us look at functions as a part of a building diagram with the knowledge that we can still figure out how much material is needed for construction.
- Section 6.1: Simple, but crucial, this section demonstrates how we apply integrals. While derivatives gave us the rate of change of the values we were examining, integrals allow us to add up a value that can be looked at as a rate of change. The most obvious use comes from rectilinear motion, where we can now begin with acceleration and work our way back to the original position.
- Section 6.2: At first, this section seems to only give us (as the title explains) the areas of regions with oddly-shaped boundaries. Consider this, however, together with the material from the previous section. If two functions represent the velocities of a pair of objects, then the area between two intersections of these functions tells us how much the distance between them is changing over this interval. The physical applications are quite extensive.
- Section 6.3/6.4: These two sections give us two methods to do the same thing. Some times only one of the methods will work, while occasionally both can be used. We are showing both methods to you together because we feel it is a better way to learn them.
- Section 6.5: A minor section with which to end our lectures, but a good one to have. From the point of view of the student, this is a nice section for exams, as there is only one formula to learn and only three types of functions that can show up for it (one of these types is constant functions, so that's even better).
|Volumes of Rotations - Volumes by Rotation: Other Axes | Link||Shai Cohen|
|Volumes of Rotations - Integration by Washers | Link||Shai Cohen|
|Volumes of Rotations - Choosing Between Washers and Shells | Link||Shai Cohen|
|Volumes of Rotations - Shells and Washers | Link||Shai Cohen|
|Volumes by Washers Example - With Limits | Link||Shai Cohen|
|Volumes by Rotation Example - With Related Rates | Link||Shai Cohen|
|Arclength - Arclength | Link||Shai Cohen|
|Arclength Example | Link||Shai Cohen|
|Velocity and Net Change - Integrals as Total Change | Link||Shai Cohen|
|Integrals as Total Change Example | Link||Shai Cohen|
|Area Between Curves - Area Between Curves | Link||Shai Cohen|
|Area Between Curves - Complications | Link||Shai Cohen|
|Area Between Curves Example - Multiple Regions | Link||Shai Cohen|
|Area Between Curves Example - The Ultimate | Link||Shai Cohen|
|Volumes of Rotations - A Slice of Calculus | Link||Shai Cohen|
|Volumes of Rotations - Volumes by Rotation: The Details | Link||Shai Cohen|