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

Error Analysis and Comparison

Error analysis in the context of numerical integration involves comparing the approximate result obtained using methods like Simpson's Rule with more precise values from other methods or computational tools. This comparison helps in understanding the accuracy of the approximation and the potential error introduced by the numerical method, ensuring that the method chosen is appropriate for the problem at hand.

Arc Length Formula

The arc length formula calculates the length of a curve defined by a function y = f(x) over an interval [a, b] using the integral L = ?[a to b] ?(1 + (dy/dx)²) dx. This integral results from summing the lengths of infinitesimal line segments along the curve and is a fundamental tool in calculus for understanding the geometry of curves.

Simpson's Rule

Simpson's Rule is a numerical method used to approximate the value of definite integrals. It works by dividing the integration interval into an even number of subintervals and approximating the function with a series of quadratic polynomials. This rule is especially useful when the exact antiderivative of the function is difficult or impossible to obtain, and it often provides a very good approximation with relatively few subintervals.

Numerical Integration

Numerical integration encompasses a range of methods used to approximate the value of definite integrals when an exact solution is impractical or impossible. Techniques such as Simpson's Rule, the trapezoidal rule, and others are employed to compute approximate values by replacing the integral with a weighted sum over a set of discrete points.

Transcript

00:01 Alright, so we have to calculate the length of the curve using simpson's formula and the value of n given to us is 10.

00:09 The function we have is y is equal to erest 2 .m, x squared and the function is defined from 0 to 2.

00:18 We have to verify the results with our formula as well, using our calculus formula.

00:24 So in both the methods, we have to, we required the first derivative of this.

00:30 So first derivative of this will be a muscle it differentiated.

00:34 So why does first derivative of this will be essentially e raised to bar minus x square and differentiation of minus x square will be minus 2 of x.

00:44 So that is the first derivative of correct.

00:47 Now let's write down the formula for length of the curve so that is l i'm calling.

00:54 So l is from integration 0 to 2 lower limit to upper limit, root or 1 plus minus 2x e.

01:05 0x2x minus x squared minus x square and horus square of that.

01:10 Okay, if you further simplify it into dx.

01:14 If you simplify it so it comes out to be 0 to 2 root hour 1 plus 4 x squared into e 2 x2 power minus 2 x squared into d x.

01:33 Okay and if we solve this integral or calculator value using calculator the value approximately we are getting is the value we are getting approximately is essentially 2 .2 8 0 526 okay so that is the value we are getting using calculator now let's use simpson's formula or simpson's method so using simpson's method in in some method, n is given as 10.

02:17 A is the low limit that is 0.

02:20 B is equal to 2.

02:21 The value of delta x we are going to get will be b minus a by n.

02:29 So 2 minus 0 divided by 10.

02:32 So the value we are getting is 1 .5.

02:35 Sorry, 1 by 5.

02:36 Okay.

02:39 Second thing is x0 will be 0...