Module 15 - Riemann Sums and the Definite Integral

Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test

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 non-negative functions it represents the exact area under f(x) and above the x-axis between x = a and x = b.

There are two features on the TI-89 that evaluate a definite integral:

1. The integral key
2. The definite integral feature of the Graph screen's Math menu

Using The Integral Key

Find the exact area under the curve f(x) = x2 and above the x-axis between x = 0 and x = 1 by using the integral key to evaluate .

• Execute NewProb and be sure that the calculator is in AUTO mode.
• Define f(x) = x2
• Clear the Home Screen by pressing  • Paste the  The syntax for finding a definite integral using the integral function is (expression, var, lower, upper)  integral function to the Edit Line by pressing  • is a second feature above the key.

• Enter [F]            This is the same value found in the previous lessons by taking the limit using both right- and left-hand rectangles.

15.3.1 Evaluate by using the definite integral key and interpret the result.

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.

• Graph y1=x2 in the window [0, 1] x [-0.2, 1]  Viewing Window Values when using f(x)dx When using the feature on the Graph screen, the window interval [xmin, xmax] must contain the interval [a, b] that denotes the interval of integration.  • Select 7: from the Math menu • When prompted, enter 0 for the lower limit and 1 for the upper limit

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].