Module 15  Riemann Sums and the Definite Integral  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 15.3: The Definite Integral  
This lesson investigates finding the definite integral The limit of a Riemann sum as the number of rectangles approaches infinity is called a definite integral. The notation used to represent a definite integral is and for nonnegative functions it represents the exact area under f(x) and above the xaxis between x = a and x = b. There are two features on the TI89 that evaluate a definite integral:
Using The Integral Key Find the exact area under the curve f(x) = x^{2} and above the xaxis between x = 0 and x = 1 by using the integral key to evaluate .
is a second feature above the key. This is the same value found in the previous lessons by taking the limit using both right and lefthand rectangles. 15.3.1 Evaluate by using the definite integral key and interpret the result. Click here for the answer. Using the Graphical Definite Integral Another way to evaluate a definite integral is by graphing the function and using the function found in the Math menu of the Graph screen.
Typing the value and pressing is a quick way to set a limit. This feature shades the area and gives a decimal approximation of the integral. The type of shading varies from one use to the next so the shading on your screen will likely be different from that in the screen shot above. 15.3.2 Find the area under the curve g(x) = 2x + 1 between x = 0 and x = 3 by using the graphical definite integral function in the Math menu of the Graph screen. Use the window [0, 3] x [2, 10]. Click here for the answer. 

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