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 problem, we have to evaluate a sigma. Okay, they kind of give us a format to

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 $5(c) .$
$$
f(x)=2 x^{2}
$$

Key Concepts

Function Evaluation in Riemann Sums

Function evaluation in the context of Riemann sums involves calculating the value of the function at specific sample points within each subinterval. These evaluated values are then multiplied by ?x and summed to approximate the integral of the function.

Riemann Sums

A Riemann sum is a method for approximating the definite integral of a function by dividing the area under the curve into small segments (rectangles) and summing their areas. This approach involves selecting points within each subinterval and multiplying the function’s value at these points by the width of the intervals.

Interval Partitioning

Interval partitioning is the process of dividing a continuous interval into smaller subintervals, each with a specified width (?x). This is crucial for numerical integration techniques like the Riemann sum, as it establishes the framework for where evaluations occur.

Delta x (Subinterval Width)

Delta x represents the constant width of each subinterval in the partition of the overall interval. In a Riemann sum, this value is multiplied by the function value at selected points in each subinterval to approximate the area under the curve.

Transcript

00:01 All right, in this problem, we have to evaluate a sigma.

00:06 Okay, they kind of give us a format to kind of follow out here and then some information about the different parts of the sigma, which i've already written down here.

00:12 Okay, and then we have a certain function we have to use.

00:14 Okay, so this video right here, we're using the function f of x equal to 2x squared.

00:21 And so we're going to find all the different terms, and then eventually have to add it together.

00:24 So for the first term here, we're doing f of x subscript 1.

00:28 What essentially means that we're just replacing any x with x subsequent 1.

00:32 And already you can see from our list right here, that that value will equal zero.

00:35 So ultimately i'm doing it.

00:36 And we also have eventually times it by point, 0 .5, because that's what change of x stands for here.

00:40 Again, in the format that i have to follow.

00:42 All right.

00:42 So kind of to condense all that, we're going to do two times x, subscript 1 squared.

00:47 Remember the x subscript 1 is equal to 0.

00:49 So i have 2 times 0 squared, and then have to times all that by 0 .5.

00:56 All right, you're going to get 0.

01:00 All right.

01:01 Next, we're doing f of x subscript 2.

01:06 That means anywhere there's an x in replacement of x subscript 2 and already you can see that's equal to 2.

01:10 So i'm just again kind of following a pattern here...

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