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

Composite Simpson's Rule

Composite Simpson's Rule is a numerical integration technique that approximates the value of a definite integral by dividing the integration interval into an even number of subintervals and applying a quadratic polynomial approximation over each pair of subintervals. This rule combines the accuracy benefits of quadratic approximations with the simplicity of computational procedures, making it effective for estimating integrals with smooth integrands.

Arc Length of a Curve

The arc length of a curve defined by a function y = f(x) is the total distance along the curve between two points. It is calculated using the integral ? from a to b of the square root of [1 + (f'(x))^2] dx, which accounts for both horizontal and vertical changes along the curve.

Numerical Integration

Numerical integration refers to the collection of methods used to approximate the value of definite integrals when an analytic solution is difficult or impossible to obtain. By replacing the integrand with simpler functions, these techniques provide estimates of the integral's value with controllable error margins, and are widely used in practical applications across science and engineering.

Transcript

00:01 Alright, so we have to calculate the length of the curve for the curve which is given as y as equals to x sine x, okay, where x is between 0 to 2 .5.

00:22 You have to calculate the length of the curve using simpson's method.

00:27 So, and we have to verify the results as well.

00:30 Okay so let's first differentiate this curve so what will be getting y -d s will be equal to the derivative of this will essentially the derivative of x will be 1 so it will be sine x plus now x will be constant and differentiation of x will be x x x so this is the derivative we have okay now the length formula is essentially length formula is integration low limit to upper limit so low limit is 0 upper limit is 2 pi under root of root hour 1 plus f dash of x square or y dash square so that is essentially square of this quantity so that will be sine of x plus x cos of x all square into d of if you simplify it so what we are going to get here is 0 to 2 pye okay root our x square cost square x square x square cost square x plus sine square x plus to sine square x we'll be getting x square cost square we will be getting 2x so it will be 2x so that will effectively become uh sine x okay we'll be getting 2x, sine x, cos x, so that will become x sine 2x.

02:20 So i'll just rough it, okay? so here it will be x sine to x and this one will also be left out so that is plus 1 into d of x.

02:51 Okay and the approximate value using calculator for this integral we are getting this essentially the approximate value we are getting using calculator is 15 .374568.

03:15 Okay, so that is the approximate value we are getting.

03:18 Now, so this value we have the length of the curve we are going using the formula.

03:23 Now we will calculate the length of curve using simpson's method.

03:28 So for simpson's method, effects.

03:32 So this this quantity will act as an fx for simpson's method okay so for simpson's method simsum method fx will be fx will be essentially root over x square cos square x plus sine square x plus x plus x sine 2x plus 1 so that will act as a fx for simpson's method okay so according to simpson method first we have to calculate delta x so delta x will be essentially final limit minus initial limit so final limit is 2 pi initial limit is 0 so 2 pi minus 0 divided by total value of n is here 10 okay 10 part we have to divided so 10 so that is simply how much pi by 5 okay so we have to calculate s 10 simpson's 10 so that will be s 10 that will be s 10 that will be delta x divided by 3 okay and that will be f of 0 plus 4 times f of 5 by 5 okay plus so on so forth last will be having 4 times plus 4 times f of 9 pi by 5 plus f of 10 5 okay so we'll calculate these values so we have to calculate these values.

06:03 Either we can make a table or otherwise if we calculate the values of s -10, let me write down all these values.

06:09 So s -10 will be actually pi by 5.

06:16 So it will be delta -x we have got as pi -by -5...

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