Find the length of the arc of the curve from point P to point Q. x^2=(y-4)^3, P(1,5), Q(8,8)

Name: Problem 20, Chapter 8: Calculus
Uploaded: 2023-05-30T07:07:54-08:00
Duration: 4 min 8 s
Channel: Ma. Theresa Alin
Description: Find the length of the arc of the curve from point P to point Q. x^2=(y-4)^3, P(1,5), Q(8,8)

Find the length of the arc of the curve from point P to...

Key Concepts

Integration Techniques

Integration techniques are critical for evaluating the definite integral derived from the arc length formula. Depending on the form of the integrand, substitutions or other methods may be employed to arrive at an exact or approximate solution for the length of the curve.

Implicit Differentiation

Implicit differentiation is the technique used to find the derivative of a variable with respect to another when the relationship between them is given implicitly by an equation. This derivative is a key component in forming the integrand for the arc length formula.

Arc Length

Arc length is a fundamental concept in calculus representing the distance along a curve. It is computed by evaluating an integral that involves the square root of the sum of one and the square of the derivative of the function or the components of a parametric curve.

Parameterization

Parameterization involves expressing a curve in terms of a parameter. This is particularly useful for curves defined implicitly because it allows one to rewrite the relationship between the variables in a form that is more amenable to differentiation and integration, which are essential steps in computing arc length.

Transcript

00:01 To determine the arc length of this curve from points p to q, we recall the arc length formula l equal to the integral from a to b of the square root of 1 plus the square of f prime of x dx or integral from c to d of the square root of 1 plus the square of f prime of y dy.

00:23 So since you can use an integral in terms of x or an integral in terms of y, we can solve or find the arc length of this curve in two ways.

00:34 Now the best way will be to rewrite this function as a function of y in terms of x.

00:40 That's because if we take the cube root both sides, we get x raised to 2 thirds equal to y minus 4, which we can simplify as y equals x raised to 2 thirds plus 4.

00:56 And if we take the derivative of that, you will get y prime equal to just 2 thirds x raised to negative 1 over 3.

01:06 If we square that, that's just y prime squared equal to 4 over 9 x raised to negative 2 over 3.

01:14 So the arc length l will be the integral from 1 to 8 in this case since we're dealing with an integral in terms of x.

01:25 So we're going to use x values from the two points.

01:29 And then we have square root of 1 plus 4 over 9 x raised to negative 2 thirds dx.

01:37 So what will happen is that it becomes integral from 1 to 8 of the square root of 1 plus 4 over 9 x raised to 2 thirds dx.

01:48 And if we combine that, it will be integral from 1 to 8 of the square root of 9 x raised to 2 thirds plus 4 over 9 x raised to 2 thirds...

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