Write the terms of ∑i=1^4 f(xi) Δx, with x1=0, x2=2, x3=4, x4=6, and Δx=0.5, for each function.

Write the terms of ∑i=1^4 f(xi) Δx, with x1=0, x2=2, x3=4,...

Key Concepts

Riemann Sum

A Riemann sum is a numerical method used to approximate the value of a definite integral. It works by partitioning the interval into smaller subintervals, evaluating the function at specific sample points within each subinterval, and then summing up the product of these function values with the width of the subintervals. This technique underpins many methods of numerical integration and forms the foundation for the formal definition of an integral.

Partitioning the Interval

Partitioning the interval involves dividing the domain of integration into subintervals of equal or varying lengths. In the context of a Riemann sum, each subinterval has an associated width (?x), and the choice of sample points (such as left-endpoint, right-endpoint, or midpoints) in these subintervals affects the sum. This partitioning is critical in breaking down the area under a curve into manageable segments for approximation.

Function Evaluation

Function evaluation in the context of Riemann sums involves substituting the chosen sample points into the function to obtain the corresponding function values. These values represent the height of the rectangles under or over the curve. Accurately computing these function values is essential for the overall accuracy of the sum that approximates the area or the integral.

Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms. In applications to Riemann sums, it is used to sum the products of function values and subinterval widths over a range of indices. This method offers clarity and brevity in expressing the cumulative process of approximating integrals and other aggregate calculations.

Transcript

00:01 All right, in this video, we're looking at evaluating a sum for a sigma.

00:04 They have a particular format we want to use, or we have to use it's f of x, subscript i times this change of x, and they already told us that what that's equal to as 0 .5.

00:14 Okay, so i have to find all the individual terms, and then i'm going to add them all up.

00:18 So the first term we're going to find is f of, oh, and i have to tell you what function.

00:23 I'm using, whoops.

00:26 Okay, so the function we're using in this video right here, we're using f of x equal to negative 2 over, x plus 1.

00:35 All right, so the first thing i want to do again is find all those individual terms in our series.

00:38 So i'm going to do f of x subscript 1.

00:41 Okay, that means wherever there's an x in the function are placing with x subscript 1.

00:44 And again, they've already told me what certain things are equal to.

00:46 And we also have the time the whole thing by 0 .5.

00:49 So i got print these right here, negative 2 over 0 plus 1, and then we're times in that by 0 .5.

00:57 Just use a calculator to help you out with that.

00:59 You'll get negative 1 for the first term.

01:01 Next, you have f of x subscript 2.

01:04 Okay, wherever it's an x, i'm replacing with x, subscript 2, and i can see that that number is going to equal to 2.

01:09 So we got print these right here, negative 2 over 2 plus 1, and it times that all by 0 .5.

01:16 Okay, so we'll give you negative 1 3.

01:19 All right, for the next one we're doing f of x subscript 3...

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