Evaluating a definite integral from a graph

The  exercise appears under the Integral calculus Math Mission. This exercise introduces the definition of a definite integral as a limit sum.

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



Strategies
Knowledge the several different numerical approximations to integrals and limit laws are encouraged to ensure success on this exercise.
 * 1) The definition of the integral is a limit of the sum of the areas of several rectangles. A rectangle is area is found by height (via the function) and width (via the length of the interval).
 * 2) When a function is increasing, L(n) underestimates and R(n) overestimates.
 * 3) When a function is decreasing, L(n) overestimates and R(n) underestimates.
 * 4) Drawing or visualizing one set of L, R and M on a picture can help to put the estimates in order efficiently.

Real-life Applications

 * 1) Geometrically an integral can be used to find area and volume formulas.
 * 2) Approximate area is useful in numerical analysis and other situations where antiderivatives are difficult or impossible.