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Handout Friday 11/14
We will start on section 4.2 today by considering a systematic way to approximate area under a curve using what is called a Riemann sum.
First, a reminder of what we are doing here. We noticed that if we are given a velocity function, we can find the change in distance by multiplying the velocity by the change it time. Okay, this only works for constant velocity. To resolve this, we will just approximate the velocity function by a sequence of constant velocities that are close to the true velocity.
It is also very helpful to picture this product of velocity by change it time as the area under the velocity curve. In fact, everything we are doing can be thought of as simply a geometry problem of finding area under a curve.
So how do we approximate this area (i.e., this change in distance)? We divide up our interval into a bunch of sub-intervals and find the area (change in distance) for each sub-interval. Then add them up.
Some notation:
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We use summation notation or Sigma notation using the Greek letter \(\Sigma\) to represent a sum. For example, \(\Sigma_{i = 1}^4 i^2 = 1 + 4 + 9 + 16\text{.}\) Try a few more examples.
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This will be useful because what we really want to do is add up a bunch of areas. So we will want to find\begin{equation*} A_1 + A_2 + A_3 + \cdots + A_n = \Sigma_{i = 1}^n A_i\text{.} \end{equation*}
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Now what is each \(A_i\text{?}\) Each is a base times a height. Let’s first think about finding the bases.
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We have some intervale \([a,b]\) that we want to find the area under (draw the picture). Suppose we divide this up into \(n\) sub-intervals. We will call each sub-interval \(\Delta x\text{.}\) What is its value? Well, \(\Delta x = \dfrac{b-a}{n}\text{.}\)
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What about the heights (velocities) for each rectangle? How tall are they? This is given to us by the function. We just need to decide what input to use to find that height.
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As an example, consider the function \(v(x) = 16 - x^2\) on the interval \([0,4]\text{.}\) Let’s first divide this interval into \(n = 2\) sub-intervals. Keep track of where each sub-interval starts and stops. Call the endpoints of the sub-intervals \(x_0, x_1, x_2\text{.}\) What inputs should we use to find the heights?
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We can choose to use the left endpoint of each sub-interval. Doing so will give us a left Riemann sum, which we would call \(L_2\text{.}\) If we choose the right endpoints, we get a right Riemann sum, denoted \(R_2\text{.}\) We could also pick the midpoint of the sub-interval, giving a middle Riemann sum, \(M_2\text{.}\) The subscript is \(n\text{,}\) so we can also ask for \(L_4\text{,}\) \(R_4\text{,}\) \(M_4\text{.}\)
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For this example, note that the following table of \(v\) values is useful:
\(x\) 0 .5 1 1.5 2 2.5 3 3.5 4 \(v(x)\) 16 15.75 15 13.75 12 9.75 7 3.75 0
