5.2B Limits of Riemann Sums and the Definite Integral

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Presentation transcript:

5.2B Limits of Riemann Sums and the Definite Integral Today, students will connect the limit of a Riemann Sum to the Definite Integral, and learn the meaning and notation for the Definite Integral.

Earlier, we learned this… …but we didn’t have much time to talk about what it means What is that thing on the right-hand side?

Since the limit of the Riemann Sum gives the area between a function and the x-axis between and it represents the same quantity as the definite integral: Now, we can see that the “dx” represents a change in x, but as n has become infinitely large, Δx has become infinitesimally small. In other words, as ,

Expanding sums: Summation properties and formulas (you are not being held responsible to know these ) constant multiple addition/subtraction

Find the right rectangular sum for the area under on the interval [5,8] using n rectangles. Then, find the limit as . This is the limit of the Riemann Sum as What did we just find?

Using NINT (or on your calculator, fnInt) Method 1: press Math 9 and enter the following parameters: Method 2: graph the function and use 2nd-Calc 7 Using NINT (or on your calculator, fnInt)

Partner Problem

The Relationship of the Integral to Area

Signed Area Consider the graph to the left. If this is the graph of the velocity of an object moving along a horizontal line over time, what does the area between the graph and the t-axis mean? The integral is defined as the “signed” area between a graph and the horizontal axis. Area above the horizontal axis is defined as positively signed area. What does this area represent in terms of the movement of the particle? Area below the horizontal axis is defined as negatively signed area. What does this area represent in terms of the movement of the particle?

This does not mean that area means anything different than it did in Geometry: The Integral in relation to Geometric area: More on Area

More practice with signed areas:

Writing an integral to model a situation:

Exploration

5.2B: p. 282: (p. 282: 16,26,30,32,33,36, 41-46) Assignment