. A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in ce

. A cross-section of an airplane wing is shown. Measurements...

Key Concepts

Partitioning Intervals

Partitioning an interval means dividing the domain of the function into smaller subintervals of equal (or sometimes unequal) widths. This step is crucial in numerical integration techniques as it sets the stage for approximating the integral by summing over these smaller segments.

Area Estimation

Area estimation in this context refers to the process of approximating the area under a curve or the total measure of a quantity, such as the cross-sectional area, using discrete data points and summing the contributions from each subinterval. This concept is fundamental in applications of numerical integration.

Numerical Integration

Numerical integration involves approximating the value of a definite integral by breaking the area under a curve into manageable parts. It is useful when dealing with functions that do not have an elementary antiderivative or when data is given at discrete points.

Midpoint Rule

The Midpoint Rule is a specific numerical integration method where the function values at the midpoints of subintervals are used to approximate the area. By calculating the value at each midpoint and multiplying by the subinterval width, one can estimate the total area under the curve.

Transcript

00:01 Hello, my name is lucas, and today we're going to be taking a look at this calculus question shown right here.

00:07 And hopefully we all will be arriving at this answer shown right here.

00:12 If you're following along the textbook, this is chapter 6, section 1, question 19.

00:19 And if you want to pause the video and write this down, feel free.

00:23 But to summarize, we have this shape right here.

00:27 It is 200 centimeters wide, and we are given the height of the height.

00:31 Of the shape or the context of the problem, the thickness of the wing at each 20 centimeter interval and those numbers are given right here.

00:41 With this information, we have to use the midpoint rule to estimate the area of this shape.

00:49 We're gonna begin with some conceptual things here.

00:52 A big part of calculus is finding the area under the curve because as we will learn, area under the curve has a lot of applications.

01:02 In calcans.

01:05 In many cases, such as this one, we're going to be given a very irregular -looking shape and we're not exactly going to know how to find the exact area of this shape.

01:17 To get around this, we're going to use approximation.

01:20 And today we're going to be learning one of these methods is called a riemensum.

01:25 And this is essentially using a bunch of rectangles whose areas we do know how to find.

01:31 It's just a simple base times height.

01:34 And to visualize this, we're going to be essentially putting a bunch of rectangles next to each other, spanning the width of this shape, adding all of their areas together to hopefully find a pretty good approximation of what the true area of this shape is.

01:55 So translating what we're given in the textbook onto paper here.

01:58 This is very poorly drawn, i'm sorry about that, but in black we have the approximate shape of what we're given.

02:08 It's 200 centimeters wide, and at each 20 centimeter partition, we have the heights that we are given in the problem.

02:20 And in green here are the rectangles we are going to be using to add together to ultimately find an approximation of, our shape.

02:31 And to do this, we have to use the midpoint rule formula, which is shown here.

02:37 It does look like a lot to look at it first, but it's essentially a fancy way of saying, we're going to be calculating the areas of a bunch of rectangles and adding them together.

02:52 In this case, another way to look at this is we can view delta x as the width of the rectangles.

03:01 Rectangles, as i wrote there...

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