The definite integral as the limit of a Riemann sum

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:


 * 1) Sort the values: This problem has several sums that approximate the area under a curve, as well as the true area. The student is expected to put these quantities in order with the smallest value on top.Tdiatloars1.png
 * 2) Find the definition of the integral: This problem presents a particular definite integral. The student is expected to find the limit sum that represents that exact integral.Tdiatloars2.png
 * 3) Find the integral that is represented: This problem presents a limit sum. This time the student is expected to figure out which definite integral is being represented.Tdiatloars3.png

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.