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,...

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 $5(c) .$
$$
f(x)=4 x-7
$$

Key Concepts

Summation Notation

Summation notation (?) is a compact mathematical way to represent the addition of a sequence of terms. In the context of Riemann sums, it is used to succinctly sum the areas of the individual rectangles (or other shapes) that approximate the total area under a curve.

Riemann Sum

A Riemann sum is a method used in calculus to approximate the area under a curve by partitioning the domain into subintervals, evaluating the function at selected sample points within these intervals, and summing the products of these function values with the corresponding subinterval widths. It is foundational for understanding definite integrals.

Partitioning of the Interval

Partitioning an interval involves dividing it into smaller subintervals, often of equal length, which makes it possible to approximate a continuous area or measure discrete values. This approach is crucial for setting up both Riemann sums and integrating functions over a specific range.

Delta x (Subinterval Width)

Delta x refers to the width of each subinterval in the partitioned interval. It is a key parameter in Riemann sums because multiplying this width by the function value at a chosen sample point in the subinterval gives the area of the corresponding rectangle.

Function Evaluation within Subintervals

Evaluating the function at specific points within each subinterval—often at endpoints, midpoints, or other specified points—is a fundamental step in approximating the area under a curve. These function values, when combined with the width of the subintervals, contribute to the overall Riemann sum.

Transcript

00:01 All right, in this video, we're looking at problems where it kind of gives us a setup for a sigma.

00:06 We have to plug a bunch of stuff in.

00:07 So for this particular set right here, we need our function to be f of x, subscript i.

00:12 And they have different values for i that we're going to plug in, or essentially what's going to equal, times the change of x.

00:18 That little delta symbol right there means change.

00:20 And again, they already gave this that information.

00:22 Now we have to know what function we're working with.

00:24 So in this video, we're going to be working with this function right here, f of x equal to 4x minus 7.

00:30 At some point, we'll have to add everything all up, but let's hang get our little pieces.

00:35 All right.

00:35 So first i'm going to find f of x subscript 1.

00:41 And that just means wherever i see an x and i replace with x subscript 1, but that essentially just means i'm plugging in zero because that's what they're telling me.

00:49 So i'm just going to kind of short hand to work a little bit.

00:52 I'm going to do 4 times 0 because that's where x, subscript 1 would go minus 7.

00:58 All right.

00:59 You'll get negative 7.

01:02 Actually, let me head and then we'll.

01:03 Also have to time just by 0 .5.

01:05 So i'm ahead and just kind of write that all out.

01:10 All right.

01:10 So again, you just plug that in all your calculator...