Given equations that define a region in the plane students are asked to find its area, the volume of the solid formed when the region is revolved around a line, and/or the region is used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since…
There is a difference between mathematics and the “real world”: In the real world you are allowed to do whatever you want, as long as there is no law against doing it; in mathematics, you cannot do something unless there is a law that says you may. A question that comes up often on the…
Integration, at its basic level, is addition. A definite integral is a sum (a Riemann sum). When you add things you get an amount of whatever you are adding: you accumulate. Here are some previous posts on this important idea that often shows up on the AP Calculus exams (usually the first free-response question!) Accumulation:…
One of the major applications of integration is to find the volumes of various solid figures. Volume of Solids with Regular Cross-sections This is where to start with volume problems. After all, solids of revolution are just a special case of solids with regular cross-sections. Volumes of Revolution Subtract the Hole from the Whole and…
Usually the first application of integration is to find the area bounded by a function and the x-axis, followed by finding the area between two functions. We begin with these problems First some calculator hints Graphing Integrals using a graphing calculator to graph functions defined by integrals Graphing Calculator Use and Definition Integrals – Exam…
Cubic Symmetry Some things are fairly obvious. For example, if you look at the graphs of a few cubic equations, you might think that each is symmetric to a point and on closer inspection the point of symmetry is the point of inflection. This is true and easy to prove. You can find the point…
Given equations that define a region in the plane students are asked to find its area and the volume of the solid formed when the region is revolved around a line or used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since 1969 on…
