Find the exact length of the curve. y=1-e^-x, 0 ⩽x ⩽2 |...

Name: Problem 18, Chapter 8: Calculus
Uploaded: 2020-05-30T15:37:02-08:00
Duration: 8 min 12 s
Channel: Rakesh Kumar Sharma
Description: Find the exact length of the curve. y=1-e^-x, 0 ⩽x ⩽2

Key Concepts

Arc Length Formula

The arc length formula is a fundamental concept in calculus used to determine the length of a curve represented by a function over an interval. For a function y = f(x), the length from x = a to x = b is given by the integral of the square root of 1 plus the square of the derivative, i.e., L = ?[a, b] ?(1 + (dy/dx)²) dx. This formula breaks the curve into infinitesimal line segments and sums their lengths to give the total curve length.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function is changing. In the context of calculating arc length, the derivative (dy/dx) is used in the arc length integrand to account for the slope of the curve. Understanding differentiation is crucial because the square of the derivative appears in the integrand inside the square root.

Integration

Integration is a key concept in calculus that deals with summing infinitesimally small quantities over a specified interval. For calculating the length of a curve, once the derivative is determined, integration is used to combine the segments of the curve's length across the entire interval, applying the arc length formula. Proficiency in integration techniques is essential to evaluate the resulting expression exactly.

Exponential Functions

Exponential functions, characterized by a constant base raised to a variable exponent, have unique properties that often lead to distinctive differentiation and integration rules. When working with exponential functions, understanding these properties facilitates the computation of their derivatives and integrals. This is particularly important when the exponential function is part of the definition of the curve whose length is being calculated.

Transcript

00:01 Alright, so let's solve the question.

00:04 We have to find the length of curve, and the curve is given as y as equals to 1 minus e .mase to power minus x.

00:15 And x is defined from 0 to 2.

00:20 0 to 2.

00:21 So now what we'll be doing is we will be differentiating the curve first.

00:26 So let's differentiate it.

00:28 So what will be getting? we will be getting y -dess and that will be differentiation of 1 will be 0 and differentiation let's put minus here differentiation of eras to bar minus x will be minus x and now differentiation of minus x will be minus 1 so it will become plus that is our first derivative for your function right so now let's integrate the function function, so the integration will be, sorry, so the length of the curve, so length of the curve, the formula is integration, lower limit to upper limit.

01:13 So, lower limit is 0, upper limit is 2, 2 tower, 1 plus, first derivative is square.

01:22 So first derivative, e raise 2 minus x, so the square of this will be e raised 2x into b of x.

01:30 Okay so now we have to integrate this integral so for that what we'll be doing is we'll assume okay i call that as i this value we have to find out so let one plus e r s to power minus 2x is equal to p square sorry okay now differentiate it on both sides so what will be getting differentiation of one will be again zero and differentiation of e rest to power minus 2x will be a s to power minus 2x into differentiation of minus minus 2x into differentiation of minus 2x will be minus 2.

02:05 Okay, and differentiation of minus 2x will be minus 2 into vx and that will become 2p d t.

02:14 Correct? this 2 2 will be canceling out.

02:18 Also from here we can find out the value of e -dase 2 power minus 2x.

02:24 So, e -raise 2 power minus 2x will be equals to 2 square minus 2.

02:30 Now let's put these values back in this integral i.

02:34 So i will be essentially, we are not putting the value of the lower and upper limit.

02:42 So i'm keeping it as it is.

02:45 So integration, the value of this thing, 1 minus e rest of 1 minus 2x will be t squared.

02:55 So this under root will be canceling out with the square...