Use Simpson's Rule with n=10 to estimate the arc length of the curve. Compare your answer with t

Use Simpson's Rule with n=10 to estimate the arc length of...

Key Concepts

Simpson's Rule

Simpson's Rule is a numerical integration technique that approximates the value of a definite integral by using quadratic polynomials to model the behavior of the integrand over subintervals. It is particularly effective when the function is smooth and when the number of intervals is even. This method combines evaluations of the function at equally spaced points to achieve a higher order of accuracy compared to methods like the trapezoidal rule.

Arc Length of a Curve

The arc length of a curve is determined by integrating the square root of the sum of 1 and the square of the derivative of the function with respect to its variable. This formula arises from approximating the curve by small linear segments, and then summing their lengths. It provides a mathematical tool to measure the length of curves, especially when they are not straight lines.

Definite Integration

Definite integration involves calculating the integral of a function over a specific interval. It is a fundamental concept in calculus that is used to compute quantities such as areas under curves, total accumulated change, and in this case, the length of a curve. Numerical methods like Simpson's Rule are often used when an analytic solution to the integral is difficult or impossible to obtain.

Derivative Computations

Calculating the derivative of a function is essential when determining the arc length of a curve, as the arc length formula involves the square of the derivative. The derivative quantifies the rate of change of the function, and its square often reveals information about the steepness of the curve. Accurate computation of the derivative is crucial for correctly setting up the integral for the arc length.

Transcript

00:02 Alright, so we have to calculate the length of the curve using simpson's method and then we have to compare the value for the length with the existing formula for length of the curve.

00:15 Okay, so the function given to us is y as equals to cube root of x where x is from 1 to 6.

00:30 Okay, so first what we'll be doing is we'll be differentiating this curve because that is anywhere so differentiation will be essentially 1 by 3 x raised to bar minus 2 by 3 so that is our first derivative of this curve okay so now the formula for length of the curve is so length of curve i'll call that as l so l will be essentially lower limit that is one upper limit that is 6 root over 1 plus first derivative derivative squared so first derivative is this so that will be essentially how much we'll be making it as 1 by 9 x x x x x x x x x x x x x x x okay so that is our length of curve using the formula right and if you calculate this value so this value comes out to be approximately using calculator this value is approximately coming out to be 5 .0 .0 .0 .000.

01:45 So this value is approximately coming out to be 5 .0.

01:49 7 409 and this is accurate to 4 digits okay so that is the value we are getting using formula but we have to calculate the length of the curve using simpson's method as well right so using simpson's method okay so for that what we'll be doing is uh we have to first see the function what is effects so the effects will be this entire thing will be ethics for simpson's formula okay so, the fx will be f of x will be actually root over 1 plus 1 by 9 x xx to power minus 4 by 3.

02:43 So that will be our fx first instance formula right.

02:47 Now we have to calculate delta x.

02:49 So delta x will be upper limit minus lower limit divided by n.

02:53 So that will be b minus a divided by n and that will be 6 minus 1 divided by 10.

03:02 Coming out to be 1 by 2.

03:05 So that is delta x we have got right.

03:07 Now, let's write the value.

03:11 Okay, so the s -10 we are going to get us essentially delta x by 3 bracket f of x0 plus f of.

03:32 So x0 will be essentially here 1.

03:35 Okay, f of x0 here will be the low limit.

03:38 So i'll directly write it as 1.

03:40 Or maybe i'll write it later on these values okay so the next value will be essentially four times f of x1 plus two times f of x2 plus so on so four till second last value will be you already know this formula so that will be four times of f of x9 plus f of x10 okay so that is how we calculate the astern values.

04:27 Now let's put these values and calculate it.

04:36 Okay.

04:37 So f of x0 is essentially 1.

04:41 So here we can write down x0 is equal to 1.

04:46 And successive values will be increased by half.

04:50 Okay? so next value will be x1, that will be 1 .5...

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