Overview
We have begun to see several reasons why we are interested in the area that lies between a given curve \(y = f(x)\) and the \(x\)-axis. The notion of a Riemann sum provides a formal symbolic way to consider approximations to this area. The notation that corresponds with Riemann sums is complicated: indeed, we will often consider expressions such as \(L_n = \sum_{i=0}^{n-1} f(x_i) \cdot \triangle x,\) and it is essential that you work to make sense of this notation and what it represents. As we progress from estimating certain areas to finding those areas exactly, we will soon start to use limits along with Riemann sums. There, too, it is essential that you recognize what a Riemann sum represents. At the core of all of this, don’t miss the fundamentally simple idea at the foundation: \(f(x_i) \cdot \triangle x\) represents the area of a certain rectangle.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Understand the basic notation for a Riemann sum. That is, when we write \(\displaystyle \sum_{i=1}^{n} f(x_i) \triangle x\), to know what this represents on a plot of \(f(x)\) and what this quantity aspires to measure.
- Understand the similarities and differences among left, right, and middle Riemann sums (\(L_n\), \(R_n\), \(M_n\)).
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- For small values of \(n\), be able to work comfortably by hand to compute \(L_n\), \(R_n\), or \(M_n\), and draw an accurate corresponding picture.
- For large values of \(n\), know how to use limits to determine the values of Riemann sums of interest. Knowing the closed-form formula for \(\sum_{i=1}^n i\) and \(\sum_{i=1}^n i^2\) may be of use.
- Understand how Riemann sums detect net-signed area between a given function and the \(x\)-axis on an interval of choice.
To prepare for class
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
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Watch the following video (by GVSUmath) which shows an example of the computation of a Riemann sum:
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Do: Experiment with this interactive applet about Riemann Sums (by Marc Renault), and use it to find \(L_{20}\) for the function and the interval discussed in the previous video. (You may change the function by entering
10*x*exp(-x)+2}in the box where the function is defined. You may change the interval where the area is estimated by clicking and dragging the blue points that are the endpoints of the interval. Finally, in the orange window to the left, you may change the number of rectangles (\(n\)) and by clicking and dragging the black point you may change the relative position (left, right, middle) of the point chosen in each subinterval.
