Find the exact length of the curve. y=√(x-x^2)+sin^-1(√(x))...

Name: Problem 16, Chapter 8: Calculus
Uploaded: 2020-05-30T14:52:39-08:00
Duration: 5 min 48 s
Channel: Rakesh Kumar Sharma
Description: Find the exact length of the curve. y=√(x-x^2)+sin^-1(√(x))

Key Concepts

Simplification of the Integrand

Once the derivative has been calculated and substituted into the formula for arc length, algebraic and trigonometric techniques are used to simplify the integrand. Recognizing patterns and identities can lead to significant simplifications, turning a seemingly complicated integral into one that is easier to evaluate exactly.

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. It provides a systematic way to differentiate a function by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function, which is essential when applying the arc length formula to complex expressions.

Differentiation of Composite Functions

Differentiating functions that are formed by the composition of two or more functions is key in finding derivatives necessary for the arc length formula. This includes using the chain rule to handle nested functions, which often appear in problems involving square roots and inverse trigonometric functions.

Arc Length Formula

This concept refers to the method for computing the length of a curve given by a function. It involves integrating the square root of one plus the square of the derivative of the function over an interval. Understanding this formula is central to solving problems related to the measurement of curve lengths in calculus.

Transcript

00:01 Alright, so we have to find the exact length of the curve, okay? and the curve given to us is y is equal to x minus x square plus sine inverse of x.

00:19 First of all, this curve is not defined for values of x, so we have to check its domain.

00:25 So if you talk about the domain of this curve, so domain will be x minus x square should be greater than equals to 0.

00:33 And if you saw it what will be getting we'll be getting x belonging to 0 to 1.

00:41 Okay? and if we get the domain of this one, so if i call that as d1, if i call that as d2, that means root of x should be 1, 2 minus 1.

00:58 And if you solve this, we'll be getting x belongs to 021.

01:04 And third thing is this x should be positive, okay? a quantity inside the and the root should be positive.

01:10 So, d3, x should be greater than equals to 0.

01:16 Okay? that means x belongs to 0 to infinity.

01:21 If we take intersection of these two, so what will be getting is the d1, intersection d2, intersection d3.

01:28 So that will be coming out to be 0, x to 4.

01:35 That means the domain of this function this given function will be from 0 to 1 okay now we have to find out the length of the curve okay so what we have to do is we have to differentiate this curve first let's differentiate it so what we'll be getting y -dess will be equals to differentiation of this first so that will be actually 1 by 2 root our x minus x square okay and now differentiation of this quantity, so that will be 1 minus 2 of x.

02:12 Okay, so that is differentiation of this and now we have to differentiate sine inverse of root x.

02:21 So, differentiation of sine inverse is 1 upon 1 minus x square, okay? and x is essentially root x here.

02:31 So this into differentiation of root x using chain root.

02:36 So, that will be essentially 1 by 2 under root of x and so that will be its differentiation...

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