Properties of integrals

The  exercise appears under the Integral calculus Math Mission. This exercise uses geometry to discover and understand some basic properties of integrals.

Types of Problems
There are two types of problems in this exercise:


 * 1) Represent the area and find the exact value: This problem presents two graphs, one which is a recognizable geometric shape and another that manipulates the shape. The student is expected to find integrals to represent both areas, and give the exact value of those areas.Propofint1.png
 * 2) Find bounds on the area: This problem presents a complicated graph of a function. The student is expected to use the minimum and maximum of the function on the interval to provide bounds on the value of the integral.Propofint2.png

Strategies
Knowledge of the various properties of area are encouraged to ensure success on this exercise.
 * 1) An integral can be split into several pieces, just like area, to find the value.
 * 2) The rectangle that completely covers a graph is an overestimate to the integral and the rectangle that is only as high as the minimum value will underestimate.

Real-life Applications

 * 1) Integrals can be used in physics to find the total distance travelled by objects.
 * 2) Integrals can be used in economics to find the total profit or cost of business ventures.
 * 3) Many times situations where one is looking for a "total" can be performed with integration.