19-21 Use Definition 2 to find an expression for the area under the graph of f as a limit. Do no

19-21 Use Definition 2 to find an expression for the area...

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the area under a curve by summing the areas of a series of rectangles. This technique involves dividing the interval of interest into subintervals, selecting a sample point within each subinterval, and summing the product of the function value at the sample point and the width of the subinterval. The sum becomes more accurate as the number of subintervals increases.

Definite Integral as a Limit of Riemann Sums

The definite integral is defined as the limit of a Riemann sum as the number of subintervals approaches infinity. This process captures the exact area under a curve over a given interval by considering infinitely many infinitesimally narrow rectangles. The formulation expresses the integral as a limit of sums, highlighting the fundamental connection between summation in the discrete case and integration in the continuous case.

Partitioning an Interval

Partitioning an interval refers to dividing the domain of the function into subintervals, typically of equal width, where the width is calculated as ?x = (b - a)/n. This concept is crucial for setting up a Riemann sum as it lays the groundwork for approximating the area under a curve by clearly defining each segment over which the function is sampled.

Selection of Sample Points

In the Riemann sum, selecting sample points (often chosen as right endpoints, left endpoints, or midpoints of each subinterval) is essential to determine the height of each approximating rectangle. The choice of sample points can affect the approximation's accuracy, but as the number of subintervals increases, the specific choice becomes less significant in the limit.

Transcript

00:01 Anytime you're asked to find the area under the graph or the area under the curve as a limit, you're really looking at a remand sum.

00:12 Okay, so when we're thinking of a remand sum, we're thinking of the form that we have summation.

00:21 And we're going to sum up a set of rectangles that represents this area.

00:26 Now, the limit is as n goes to infinity.

00:31 And the reason why n is going to infinity is because we have n rectangles.

00:37 And if we want the exact area under the curve, then we will make n go to infinity because that means we want infinite number of rectangles.

00:50 And there's really no way to perfectly add up those infinite rectangles because we can't count them, but we could represent it and eventually with a limit and with that representation of a limit, we actually could figure it out.

01:07 So we start with, we're going to start with i from 1 to n.

01:13 And that represents the number of rectangles.

01:16 And again, we're going to have how many rectangles? we're going to have infinite rectangles.

01:20 Now, since the f of x, just for a second, let's write down over here, the f of x is 2x over x over x squared.

01:27 Plus one.

01:28 And when we think of the area of a rectangle, let's calculate that as well.

01:34 When we think of area of a rectangle, we're thinking of length times width.

01:39 Now let's think of the area under a curve with rectangles.

01:43 So if we divide this up by rectangles, here we go.

01:47 It looks something like this, and we're looking for these areas right here.

01:55 And let's say we're going from the right side.

01:57 We usually do this as a right hand sum.

02:01 So we're going from the right side.

02:03 So i'm drawing these all from the right side.

02:05 So the widths are all the same.

02:09 Let's call these changes of x.

02:11 These are our widths.

02:14 And then the heights are f of x's.

02:21 The heights are f of x sub 1 and then our f of x sub 2 and so on at each rectangle...

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