Trapezoid rule

The  exercise appears under the Integral calculus Math Mission. This exercise explores the trapezoid rule for approximating the area under a curve.

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


 * 1) Determine how the estimate is: This problem states several approximations to a function. The student is asked to determine if the estimates are overestimates or underestimates to the true value.Traprule1.png
 * 2) Find the representation of the sum: This problem presents a particular function and a number of partitions. The student is expected to use the trapezoid rule to find an approximation to the area under the function.Traprule2.png
 * 3) Put the objects in order: This problem presents several approximations and the exact area under a particular curve. The student is asked to but the quantities in increasing numerical order.Traprule3.png

Strategies
Knowledge the several different numerical approximations to integrals are encouraged to ensure success on this exercise.
 * 1) The L(n) is a left-endpoint estimate, R(n) is right, M(n) is midpoint and T(n) is trapezoids. The n always represents how many subdivisions are used.
 * 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) When a function is concave up, T(n) overestimates.
 * 5) When a function is concave down, T(n) underestimates.
 * 6) Drawing or visualizing one set of L, R and T 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.