Use the arc length formula to find the length of the curve y=√(2-x^2), 0 ⩽x ⩽1 . Check your answ

Use the arc length formula to find the length of the curve...

Key Concepts

Definite Integration

Definite integration involves evaluating the integral of a function over a specific interval. It allows us to calculate the exact arc length by integrating the expression ?(1 + (dy/dx)²) from the start to the end of the interval, yielding a numerical value for the length of the curve.

Geometric Interpretation

Recognizing that a curve represents part of a familiar geometric shape, such as a circle, can provide a way to verify the result obtained through integration. Knowledge of geometric properties, like the circumference of a circle, can simplify computations and serve as a check against the calculated arc length.

Arc Length Formula

This is a method in calculus used to determine the length of a curve defined by a function. The formula, typically written as L = ? from a to b ?(1 + (dy/dx)²) dx for functions expressed in the form y = f(x), requires calculating the derivative of the function and integrating over the interval of interest.

Differentiation

Differentiation is the process of computing the derivative of a function, which measures the rate at which the function changes. In the context of finding an arc length, the derivative is essential because its square is used under the square root in the integrand of the arc length formula.

Trigonometric Substitution

This technique is used to simplify integrals that involve square roots of quadratic expressions. Trigonometric substitution converts algebraic expressions into trigonometric forms, which are often easier to integrate, and is particularly useful when applying the arc length formula to curves that result in complex integrals.

Transcript

00:01 Alright, so this question we have to find out the arc length, okay? arc length for the function y is 2 to root on 2 minus x square where x is from 0 to 1.

00:19 Okay? let's first differentiate this function so calculate the first genuity for this function.

00:27 So, y is given, so y -dice will be essentially how much? if you differentiate this, third will be getting, we'll be getting 1.

00:34 By 2 in division root our 2 minus x square in differentiation of minus x squared will be minus 2 x 2 will cancel out so essentially the first derivative will be rating as minus x divided by root over 2 minus x squared so that is first derivative for this function line okay now we required arc length so the formula for arc length is the formula for arc length is then goes to integration a to b low limit to upper limit two tower 1 plus f -dose of x square of that into d of x okay so let's put these values so the lower limit here is the lower limit here is essentially is 0 you see the low limit this is 0 and upper limit is 1 so low limit is 0 upper limit is 1 and we'll be having root tower our function is 1 1.

01:45 Plus x squared divided by 2 minus x squared into d of x okay.

01:54 Now if we further simplify it, so what will be getting it? 0 to 1, integration 0 to 1.

02:01 And if you simplify it so what will be getting is, let's take l -cm inside.

02:08 So 2 minus x square plus x squared divided by 2 minus x square into d x so that is what will be correct now this x square will be cancelled by this x squared so we will be in the numerator we have only root 2 so let's take out that root 2 outside of this integral so we'll be getting root 2 integration 0 to 1 bx divided y root 2 square minus x square okay and end the root of that correct so we have to find out integration of that we have a formula for that and the formula is integration d x divided by root over a square minus x square as it goes to side universe of x by a plus c okay so we'll be using the formula so what we'll be getting here is root 2 so the integration what the integration we are going to make this root 2 into 2, sine inverse of xx by root 2 and the limits here will be upper limit will be 1 and lower limit will be 0.

03:39 If you put these values to what will be getting as root 2, sine inverse of 1 by root 2.

03:48 We will put the upper limit that is essentially, we will put, we will be putting the upper limit that is essentially 1 and the lower limit that is 0.

04:02 So, upper limit we have put as 1, okay? and low limit will put 0, so sine inverse of 0 will be 0.

04:12 And if we simplify it, so what will be getting root 2 into sine nodes of 1 by root 2 will be pi by 4? so the answer we are going to get here is pi by root 2, sorry, pi root 2 divided by 4.

04:27 That is the answer we will be getting it.

04:29 Okay, now in the question they have also given that we have to verify the results, okay, as this function is essentially a part of a circle or a arc of a circle.

04:40 So we have to calculate the arc length.

04:42 So for that, what we will be doing is we will be essentially, let's make a circle.

04:51 The center of the circle will be 0 .0.

04:57 How we can do, ensure that the center of the circle is 0 .0? what you can do is essentially you can square on both side and take x squared down on this side.

05:08 So the equation of the circle will be x squared plus size.

05:10 Is equal to 2.

05:12 Now, that is y square.

05:17 So when x is equal to 0, the value we are getting is y is equal to root 2...

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