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The Mean Value Theorem for Integrals:
Let be a differentiable and continuous function on . Then the average value is given as: .
Basic Integration Rules:
Upper and Lower Riemann Sums:
(Lower Riemann Sum) where for [a,b] and where n is the number of subintervals.
(Upper Riemann Sums) where for [a,b] and where n is the number of subintervals.
Rapid Study Kit for "Title":
Core Concept Tutorial
Problem Solving Drill
Review Cheat Sheet
"Title" Tutorial Summary :
This tutorial introduces basic concepts of integration and builds the foundation for the introduction of the concept of finding the area of a region using Riemann sums and consequently integration. The First and Second Fundamental Theorem of Calculus are presented with the aid of examples. All theorems are discussed with definitions, proof and examples.
The Mean Value Theorem as it corresponds to intervals is mentioned in this tutorial with examples. The concept of using a change of variables to evaluate an indefinite or definite integral is given here with an innovative strategy. Determining how to integrate an even or odd function is given with a series of scenarios.
Specific Tutorial Features:
Detailed examples to show important concepts such as the Mean Value Theorem for Integrals..
Step by step examples are shown to handle the concept of integrating a definite or an indefinite integral.
A classification of what functions are odd or even is given in the form of a table,
Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
Definition slides introduce terms as they are needed.
Visual representation of concepts
Animated examples—worked out step by step
A concise summary is given at the conclusion of the tutorial.
"Title" Topic List:
General solution of a differential equation Indefinite integral notation for antiderivatives Basic integration rules to find antidenivatives Sigma notation to write an evaluate a sum Area of a plane region Area of a plane region using limits The definition of a Riemann Sum Evaluating a definite integral using limits Evaluating a definite integral using properties of definite integrals The Fundamental Theorem of Calculus and the definite integral The Mean Value of Integrals The average value of a function over a closed interval The Second Fundamental Theorem of Calculus Using pattern recognition to evaluate an indefinite integral Use a change of variables to evaluate an indefinite integral Using the General Power Rule for integration to evaluate an indefinite integral Use a change of variables to evaluate a definite integral Evaluate a definite integral involving an even or odd function