Applications of Integration
Lessons in this chapter
About Applications of Integration
Applications of Integration closes the course by using the definite integral to measure real areas and volumes.
Area Under a Curve Using Definite Integrals and Total Area and Regions Below the Axis cover finding the exact area under a graph, including regions that dip below the x-axis.
Areas of Symmetric Regions and Area Between Two Curves extend area calculations to symmetric shapes and to the region trapped between two different curves.
Areas of Composite Regions combines multiple curves and boundaries into one calculation, and Volumes and Solids of Revolution closes the chapter by finding the volume of a 3D solid formed by spinning a 2D region around an axis.
Six lessons, each a worksheet worked in the page, run from Area Under a Curve Using Definite Integrals through Area Between Two Curves to Volumes and Solids of Revolution. The problems are scored as you answer them, so an integral set up with the lower curve on top is caught at that question. Free to practice online.
Frequently asked questions
Area Under a Curve Using Definite Integrals turns the integral into an area, and Total Area and Regions Below the Axis deals with the negative value a region below the axis contributes. Areas of Symmetric Regions uses symmetry to halve the work, Area Between Two Curves integrates the difference of an upper and a lower function, and Areas of Composite Regions splits an interval where those two swap places. Volumes and Solids of Revolution closes by spinning a region around an axis and integrating its cross-sections. This is Unit 8 of the AP Calculus AB course framework, Applications of Integration, beyond anything the Common Core covers.
A definite integral directly gives the exact area between a curve and the x-axis over a given interval.
A region below the x-axis contributes a negative value to a definite integral, so finding total (unsigned) area requires either taking absolute values or splitting the integral where the curve crosses the axis.
A symmetric region’s area can often be found by calculating just half of it and doubling, using the symmetry to simplify the integral.
The area between two curves is found by integrating the difference between the upper and lower function over the interval where they bound a region.
A composite region combines several curves or boundaries, often requiring the interval to be split into pieces, each with its own upper and lower function, before adding the results.