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.

step 1 All right, in this video, we're writing out the terms of a series. I'm going to for this particu

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

Write the terms of $\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,$ with $x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,$ and $\Delta x=0.5,$ for each function. Evaluate the sum. See Example $5(c) .$
$$
f(x)=6+2 x
$$

Key Concepts

Riemann Sum

A Riemann sum is a method used to approximate the definite integral of a function over an interval by summing the areas of rectangles with heights given by the function's value at selected points within each subinterval.

Partition of an Interval

This concept involves dividing the overall interval into smaller subintervals. The chosen points within these subintervals, which can be endpoints, midpoints, or other sample points, are used to evaluate the function in order to estimate the area under the curve.

Delta x (Subinterval Width)

Delta x represents the width of each subinterval formed by partitioning the interval. It is a critical component because it determines the base length of each rectangle in the Riemann sum, and uniformly multiplying it by the function's value at the sample point gives the approximate area of that segment.

Function Evaluation at Sample Points

This concept refers to substituting a chosen sample point from each subinterval into the function at hand. The value obtained is then multiplied by Delta x to contribute to the overall Riemann sum, estimating the area under the curve for that segment.

Sigma (Summation) Notation

Sigma notation is a concise way to represent the sum of a sequence of terms. In the context of Riemann sums, it is used to sum the individual products of function values and Delta x across all subintervals, thus encapsulating the entire approximation process.

Transcript

00:01 All right, in this video, we're writing out the terms of a series.

00:04 I'm going to for this particular kind of format of a function right here.

00:07 I'll have to write down the function a second that we're working with, and they give us different values that we have to use in our computation.

00:14 So for this video, we're looking at this function right here.

00:17 It's going to be f of x equal to 6 plus 2x.

00:22 All right, and then we have to again figure out all the terms and then eventually have them together.

00:26 So the first time we do is find f of, and then our first part of our evaluation, x, is the one right here.

00:32 So we're doing x subscript 1.

00:34 Okay, that means where there's an x in that function we're placing with the x subscript 1.

00:38 And again, i have already what those numbers equal to.

00:41 And now i've been, and if you look right here, we're timesing it by point, i'm sorry, we're timesing it by 0 .5 because that's what the change of x stands for.

00:48 They've already told us that.

00:49 All right, so i'm doing print the seed right here, six plus two times x subscript 1 because that's what we're replacing.

00:56 Okay, i already know that 0.

00:58 And then we have the times the whole thing by 0 .5.

01:02 All right, if you work that out, you'll get three.

01:05 So that's the first term.

01:07 Okay, next.

01:08 So i kind of shorthanded the work a little bit instead of doing it a bunch of steps.

01:11 So we got f, x, subscript 2.

01:14 Okay, wherever it's an x in that function, replacing with x subscript 2, okay, i already know that's equal to 2 for my list right here...