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

Key Concepts

Arc Length Formula

The arc length formula is a fundamental concept in calculus used to determine the length of a curve described by a function. The formula involves integrating the square root of 1 plus the square of the derivative of the function with respect to its independent variable over the given interval. This approach converts a geometric problem into an analytic procedure, allowing one to compute the length of curves that are not easily measured by straight-line segments.

Differentiation and Integration Techniques

To apply the arc length formula, one must differentiate the function to find its derivative and then simplify the expression under the square root sign in the integrand. This process often involves algebraic manipulation and the use of trigonometric substitutions when integrating. Recognizing patterns that simplify the integration process is key to efficiently computing the arc length.

Circle Geometry

The connection to circle geometry is crucial in understanding the geometric context of the problem. When a curve represents a portion of a circle, its length can be compared or directly determined using known properties of circles, such as the circumference and the fraction of the circle represented by the curve. This geometric insight serves as a useful check for the result obtained via calculus.

Transcript

00:01 To find the arc length of the curve y that's equal to the square root of 4 minus x squared on the closed interval 0 to 2, we recall the arc length formula defined by l which is equal to the integral from a to b of the square root of 1 plus the square of f prime of x dx.

00:16 Now, our function here is y equal to the square root of 4 minus x squared, so f prime of x will be equal to the derivative of that which is one half of 4 minus x squared raised to negative one half times negative 2x which we can simplify into negative x over the square root of 4 minus x squared.

00:46 So if we square this, f prime of x squared will equal x squared over 4 minus x squared.

00:56 Thus, the arc length l will be equal to the integral from 0 to 2 of the square root of 1 plus x squared over 4 minus x squared dx.

01:08 And if we simplify that further, it's gonna give us the integral from 0 to 2 of the square root of 4 minus x squared plus x squared over 4 minus x squared dx.

01:19 It's gonna give us the integral from 0 to 2 of the square root of 4 over 4 minus x squared dx.

01:26 Now, using properties of radicals, this can be rewritten further into the integral from 0 to 2 of 2 over square root of 4 minus x squared dx.

01:39 And then we can bring the 2 in front of the integral, and we can use this basic antiderivative here.

01:49 That is, if we have the integral of 1 over square root of, say, a squared minus u squared du, this is equal to sine inverse of u over a plus c.

02:02 So from here, we should get 2 integral from 0 to 2 of 1 over square root of 2 squared minus x squared.

02:12 So that'll be 2 sine inverse of x over 2, and then evaluated from 0 to 2...

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