Find the exact length of the curve. x=(1)/(3) √(y)(y-3), 1 ⩽y ⩽9

Name: Problem 11, Chapter 7: Essential Calculus Early Transcendentals
Uploaded: 2020-05-30T09:36:43-08:00
Duration: 5 min 59 s
Channel: Rakesh Kumar Sharma
Description: Find the exact length of the curve. x=(1)/(3) √(y)(y-3), 1 ⩽y ⩽9

step 1 And this is defined for Y firm 1 to 9. So for that, what we have to do is we have to first calcu

Find the exact length of the curve. x=(1)/(3) √(y)(y-3), 1...

Key Concepts

Arc Length Formula

The arc length formula provides a method for computing the length of a curve by integrating the square root of one plus the square of the derivative of the function with respect to its variable. For a curve given as x = f(y) or y = g(x), the formula involves integrating ?(1 + (dx/dy)²) or ?(1 + (dy/dx)²) over the specified bounds.

Differentiation

Differentiation is used to calculate the rate of change of the function with respect to its variable, which is essential in finding the derivative needed for the arc length formula. When the function involves products or compositions, applying the product rule and chain rule correctly is crucial to obtain the correct derivative.

Definite Integration

Definite integration is the process of calculating the accumulation of quantities, such as the length of the curve over a specified interval. In the context of arc length, the definite integral evaluates the total length of the curve by integrating the arc length differential, using appropriate limits corresponding to the interval over which the curve is defined.

Algebraic Simplification

Algebraic simplification involves reducing complex expressions to simpler forms to make them easier to integrate. This can include expanding squared terms, factoring, or applying substitutions, which are often necessary steps when dealing with the integrals that arise in arc length computations.

Transcript

00:01 And this is defined for y firm 1 to 9.

00:08 So for that, what we have to do is we have to first calculate the first derivative.

00:12 So first derivative, so here we'll differentiate with respect to y.

00:17 So what we'll be getting is, we'll be getting, we'll differentiate in the first term.

00:20 So root y into y, so that will be the 5 is 2 by 2.

00:24 So if we differentiate that, so what will be getting 1 by 3 by 2 into root of y minus.

00:31 Minus now if i multiply root y with this 3, outset this entire time with this 3, so what will be 3 will cancel out? so i have to differentiate only root y.

00:42 So the value of 4 or there will be essentially 1 upon 2 root of y.

00:49 So that is what i will be getting.

00:50 This 3 will cancel out.

00:53 So the first derivative we will be getting is 1 by 2 into root y minus 1 upon root y.

01:01 So, that is the first derivative.

01:04 So let's calculate the length of the curve.

01:06 So length of the curve will be equals to 1 to 9 root or 1 plus 1 by 4.

01:24 So first derivative is this, the square of that will be 1 by 4 hour root of y minus 1 by root of y and whole square of that.

01:36 And that is divide.

01:38 Okay? now if we further simplify it, so what will be getting integration 1 to 9, 1 plus, so we'll square it so we will be getting y by 4 plus y plus 1 by 4 into y minus 2 root y into 1 by root y.

02:09 So essentially that will be and that will multiply by 1 by 4.

02:12 So that will be actually 2 by 4.

02:16 So that will be 1 by 2, you will be getting into d of y.

02:21 If we further simplify it, so 1 by 9 and root over, you see 1 minus 1 by 2.

02:31 So essentially it will become plus 1 by 2.

02:34 And if we see it, it will plus 1 by 2 here.

02:38 So the thing entire this under root will become y by 4 plus 1 by 4 plus 1 by 2.

02:45 And that is essentially nothing but that will be 1 by 4 into root y plus 1 by root of y whole square of that into d of 1.

03:03 Now take everything out so this square will be this square will be cancelled with this under root and 1 by 4 under root will be 1 by 2 and we will take that outside of that integrals...

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