Module 17  Riemann Sums and the Definite Integral  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 17.3: The Definite Integral  
This lesson investigates finding the definite integral, a basic construct of calculus, using the TI83. For a given function over a given xinterval, the limit of the Riemann sum as the number of rectangles approaches infinity is called a definite integral. The notation used to represent a definite integral is . If a < b and f(x) is nonnegative on the interval [a, b], the integral represents the exact area under the curve y = f(x) above the xaxis between x = a and x = b. The values of a and b are called the limits of integration. There are two features on the TI83 that approximate a definite integral:
Both do the same thing and will always give the same answer. Using the fnInt Command The area under the curve f(x) = x^{2} above the xaxis between x = 0 and x = 1 is given by . Use the fnInt command on the Home screen to approximate this integral by following the procedure below.
The exact value is 1/3. The fnInt command gives an approximation, and one cannot always count on it being all that accurate. However, for "typical" functions on reasonably sized intervals, it will be. 17.3.1 Approximate using the fnInt command 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 Definite Integral function, , which is found in the CALC menu of the Graph screen.
The Definite Integral command of the Graph screen's CALC menu shades the area and gives a decimal approximation of the integral. 17.3.2 Find the area under the curve g(x) = 2x + 1 between x = 0 and x = 3 by using the Definite Integral command in the CALC menu of the Graph screen. Use the window [0, 3, 1] x [2, 10, 1]. Click here for the answer. 

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