Functions defined by integrals

The  exercise appears under the Integral calculus Math Mission. This exercise defines a function in terms of an integral and explores how this relates to the graph.

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


 * 1) Order the values: This problem presents a graph and names a function F based on the graph. The student is expected to take several values of F and place them in numerical order from smallest to largest.Fdbi1.png
 * 2) Find where the value occurs: This problem presents a graph and a function defined by the graph. The student is expected to find the inputs that cause a given value to be the output of the function.Fdbi2.png

Strategies
Knowledge of integral properties and the meaning of the derivative are encouraged to ensure success on this exercise.
 * 1) Since the function is defined as an integral, the graph represents the derivative of F. This can be used to figure out when F is increasing, decreasing, etc.
 * 2) The figures in the graphs tend to be piece-wise linear so area can be found with rectangles and triangles.
 * 3) One spot where an integral evaluates to zero is when the lower and upper bounds on the integral are identical.

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.