Find the exact length of the curve. y=(1)/(4) x^2-(1)/(2) lnx, 1 ⩽x ⩽2

Name: Problem 15, Chapter 8: Calculus
Uploaded: 2020-05-30T09:52:41-08:00
Duration: 4 min 37 s
Channel: Rakesh Kumar Sharma
Description: Find the exact length of the curve. y=(1)/(4) x^2-(1)/(2) lnx, 1 ⩽x ⩽2

Find the exact length of the curve. y=(1)/(4) x^2-(1)/(2)...

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Key Concepts

Definite Integration

Definite integration is applied to accumulate the infinitesimal lengths computed by the arc length formula over the specified interval. This process results in the exact length of the curve between the given bounds, making it a crucial step in solving problems related to arc length.

Algebraic Simplification

Algebraic simplification is key in handling the expression under the square root in the arc length formula. Simplifying this expression, often by completing the square or factoring, can transform a challenging integral into one that is easier to evaluate, sometimes even reducing it to a standard form.

Arc Length

The arc length of a curve is determined using the formula that involves integrating the square root of 1 plus the square of the derivative (dy/dx) over the given interval. This concept is fundamental in calculus for finding the distance along a continuously curved path.

Differentiation

Differentiation is used to compute the derivative of the function, which represents the slope of the tangent line at any point on the curve. This derivative is squared and incorporated into the arc length formula, making its accurate computation essential for finding the correct arc length.

Transcript

00:02 Alright, so let's hold the question we have to find the length of the curve, okay? and the function given to us is essentially y is equals to x square by 4 minus lmx by 2.

00:27 Where x is from 1 to 2.

00:30 Okay, so for that first we'll differentiate this curve and get the first derivative.

00:38 So first derivative will be 2x divided by 4 minus differentiation of lnx will be 1x, 1 by 2x.

00:48 So here we will be adding 2.

00:50 So the first derivative here we are going to have is 1 by 2 into x minus 1 by x.

00:57 So that is the first derivative we get.

00:59 Now get the length of the curve.

01:02 Okay.

01:03 So then now will be integration 1 to 2 root of a 1.

01:15 Plus first derivative square and that is first derivative so that will be essentially 1 by 4 x minus 1 by x whole square into dfx.

01:29 If you further simplify it so what will be getting is integration 1 to 2.

01:37 If you simplify it what we are going to get is 1 plus x square by 4, i am multiplying 1 by 4 as well plus 1 by 4 x square, okay, minus 2x into 1x x .0.

02:04 X will cancel out so we will be getting 2 by 4.

02:06 So that will be 1 by 2 into d of x.

02:10 Let's further simplify it.

02:11 So we will be getting integration 1 by 2 to 2 2 tower...

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