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 asked to evaluate the sum of a sigma. They have a particular for

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 technique used to approximate the area under a curve by dividing the interval into smaller subintervals, evaluating the function at selected points within these subintervals, and summing up the products of these function values with the corresponding subinterval widths. This method is a fundamental concept in integral calculus and provides the basis for the definition of the definite integral.

Partitioning of Intervals

Partitioning involves dividing the overall interval of interest into smaller, equally or unequally sized subintervals. The choice of these subintervals and their widths plays a crucial role in approximating the area under a curve, as each subinterval provides a segment over which the function contributes to the overall sum.

Function Evaluation at Sample Points

In the context of Riemann sums, the function under consideration is evaluated at specific sample points within each subinterval. These evaluations are then used to determine the contribution of each subinterval to the overall area approximation when multiplied by the subinterval's width.

Summation Notation

Summation notation provides a compact and standardized way to represent the addition of multiple terms, such as the products of function values and subinterval widths in a Riemann sum. This notation efficiently encapsulates the process of summing a series of terms, which is essential for expressing and manipulating the approximation of integrals.

Transcript

00:01 All right, in this video, we're asked to evaluate the sum of a sigma.

00:03 They have a particular format that we need to follow.

00:06 It's f of x, subscript i times the change in x.

00:09 And again, i've already told me what that's equal to and a bunch of pieces of information i need to know as well.

00:14 Okay, so i've already written that out.

00:15 Okay, i do have to tell you what function we're working with for this one.

00:17 We're doing f of x equal to 5 over 2x minus 1.

00:24 So the first thing to do is find the individual pieces of the series and then eventually add that all up.

00:29 So the first thing i'm going to do is find f of x, x, subscript 1.

00:33 And that just means anywhere there's an x in that function are replacing with x subscript 1.

00:37 And again, i already know what certain things are equal to.

00:39 So if i plug an x, subccript 1, that's actually equal to the number of 0.

00:43 And just don't forget, after we do that, we have the times it all by 0 .5, because you're multiplying that piece by that change of x, which is 0.

00:49 All right.

00:50 So i'm going to do print these right here, 5 over 2 times, again, where that x -substrip 1 would go.

00:55 I'm replace that with a 0 minus 1 and times that by 0 .5.

01:00 Just be careful for this one.

01:01 When you plug it in, there's a lot of little things going on.

01:02 With fractions and stuff that you type it incorrectly in your calculator.

01:06 Okay.

01:07 All right, if you do this, you should get negative five halves.

01:11 All right, for the next one, we're doing an f of x, of x, subscript 2.

01:14 That means wherever there's an x in replacement, x, subscript 2, but i already can see from my list right here, it's a 2.

01:19 Right, so we're do parentheses, 5 over 2 times 2 minus 1, and we're at the times that all by 0 .5...