Find the exact length of the curve. x=(y^4)/(8)+(1)/(4 y^2), 1 ⩽y ⩽2

Name: Problem 10, Chapter 8: Calculus
Uploaded: 2020-05-30T09:25:58-08:00
Duration: 6 min 40 s
Channel: Rakesh Kumar Sharma
Description: Find the exact length of the curve. x=(y^4)/(8)+(1)/(4 y^2), 1 ⩽y ⩽2

Find the exact length of the curve. x=(y^4)/(8)+(1)/(4 y^2),...

Key Concepts

Arc Length Formula

The arc length formula is a fundamental concept in calculus that provides a method for computing the distance along a curve. When a curve is defined by an equation in terms of one variable (for example, x as a function of y), the formula involves integrating the square root of one plus the square of the derivative of the curve with respect to that variable. This concept is crucial when transitioning from a geometric description of curves to a quantitative measurement of their length.

Differentiation with Respect to the Independent Variable

Differentiation is the process of computing the rate at which a function changes with respect to its independent variable. In the context of finding the arc length of a curve defined by x as a function of y, it involves calculating the derivative dx/dy. This derivative is used in the arc length formula, and accurately computing it is essential for setting up the correct integral for the curve's length.

Integration Techniques

Integration is used to compute the accumulated quantity—in this case, the total distance along the curve—by adding up infinitesimal segments. After forming the integrand from the arc length formula, various integration techniques such as substitution and algebraic manipulation may be applied to simplify and evaluate the integral exactly. Mastery of these techniques is important for solving a wide range of problems in calculus involving continuous accumulation.

Transcript

00:01 Alright, so we have to find out the exact length of the curve, okay? so the exact length of the curve and the curve is given as x as he goes to y is to power 4 divided by 8 plus 1 by 4 y square where y is between 1 to 2.

00:24 So let's first get its first derivative so we'll differentiate it with respect to y so we'll got it first derivative.

00:31 So x -dust will be 4 y cube divided it.

00:34 By 8, okay, and differentiation of this will be essentially plus 4, sorry, okay, so this will be 1 by 4, okay, into differentiation of y square, 1 upon y square, so take y square into numerators, so minus 2, that will be minus 2 divided by y q.

01:12 This 2 will be cancelled out with this, 2 will be getting, here it will be canceling, so we will be getting 2.

01:18 Now we can take 1 by 2 common so 1 by 2 we have taken common and what will be left with 5 .2 per cube minus 1 by 5 plus 2.

01:28 So that is what we will be getting.

01:30 Now simplify, so that is the first derivative you got.

01:33 Now get the length of the count.

01:36 So formula for length curve which is 1 to integration 1 to 2 root or 1 plus 1 plus 1 plus 1 plus f -dase of square.

01:54 Or first derivative square.

01:58 So, square of this.

01:59 So this will be 1 by 4 into y cube minus 1 by y cube square into d of y.

02:09 If further simplify it, so what will be getting 1 to 2, open it.

02:16 So we will be getting 1 plus y -dhast 2 palm 6 divided by 4, okay, plus 1 by 1 by 1, 4 yq, y rest to par 6.

02:42 Essentially it will be a q, a square plus b square minus 2 a b and everything will be multiplied by 1 by 4.

02:49 Okay, so that will be 1 by 4 multiplied by minus 2.

02:54 So it will become 1 by 2.

02:56 If you further simplify it, so what will be getting 1 by 2? you see here, this is y rest to par 6 we have got.

03:09 Okay? okay.

03:13 And this is essentially 4 y as to 4 minus 6 minus y.

03:24 Okay.

03:25 So what we can do here essentially if you further simplify so we will be getting it as y as to pass 6 by 4 plus 1 upon 4, y is to 4 x 2 1 minus 1.

03:43 So we will be getting plus half...

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