Finding Distance Using Two Methods In Exercises 5 and 6, find the distance between the points us

Finding Distance Using Two Methods In Exercises 5 and 6,...

Key Concepts

Distance Formula

This formula is used in coordinate geometry to find the straight-line distance between two points. It is based on the Pythagorean theorem, which relates the sides of a right triangle. The formula calculates the distance as the square root of the sum of the squares of the differences between corresponding coordinates.

Arc Length (Integration Method)

Arc length in calculus is determined by integrating the length of a small segment of a curve over an interval. When the curve is a straight line, the integral simplifies to the distance formula, but this method also applies to more complex curves. It involves parametrizing the line or curve and integrating the square root of the sum of the squares of the derivatives of the parametric equations.

Transcript

00:01 Okay, so this problem asks us to find the length or distance between these two points, negative 2 -2 and 4 -96, and doing this two ways.

00:12 One using the standard distance formula and the other one using the arc length formula with an integral.

00:20 Okay, so let's do the distance formula first.

00:30 So the distance between two points, which is just a straight line between them, is given by the square root of x2 minus x1 squared plus y2 minus y1 squared.

00:55 Okay.

00:58 So in this case x2 will be negative 6, y2, oops, sorry, x2 will be 4, y2 will be negative 6, x1, x1, x2 will be 4, y2 will be negative 6, x1, x1 will be negative 2 and y1 will be 2.

01:22 So then this just becomes square root 4 minus negative 2 squared plus negative 6 minus 2 squared.

01:43 So this is 6 squared or 36 plus this will be negative 8 squared, which is 64, all under the radical.

01:55 This is just square root of 100 or 10.

02:01 Excellent.

02:03 So let's see if we do it the other way, if we can get the same result.

02:08 So here we're using the arc length formula.

02:14 We're just going to take the integral of the square root of one plus, and this will be the derivative, but i'm going to write it like this, change of y over change of x, dx.

02:30 And it makes sense to write that instead of, for example, f prime of x.

02:36 Since in this case we are dealing with a straight line, you could think about it either way...

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