# Trapezoidal rule to estimate arc length error

• Zack K
In summary: Wow that actually makes so much sense. My biggest issue is once I got the second derivative, I got this large function which seemed annoying to evaluate with x=5 (given that this question could be on an exam). Thank you.
Zack K
Homework Statement
State the integral (do NOT evaluate) to compute the arc length between x = 1 and x = 5 for the function ##y=\frac{1}{x^2}## (already done)

How many intervals are required to numerically compute the integral in to an accuracy of ##10^{-3}## using the trapezoidal rule?
Relevant Equations
##L=\int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx##

##E_T\leq\frac{(b-a)^3}{12n^2}[max |f^{(2)}(x)|]##
I got the first part of it down, $$L=\int_1^5 \sqrt{1+(\frac{1}{x^2})}dx$$

I just want to know if it's right to make your ##f(x)=\sqrt{1+\frac{1}{x^2}}## then compute it's second derivative and find it's max value, for the trapezoidal error formula.

Zack K said:
I got the first part of it down, $$L=\int_1^5 \sqrt{1+(\frac{1}{x^4})}dx$$
Your integrand isn't right. What is ##\frac d{dx}\frac 1 {x^2}##? In your integrand above, it looks like you just squared ##\frac 1 {x^2}##, and forgot to find its derivative.
Zack K said:
I just want to know if it's right to make your ##f(x)=\sqrt{1+\frac{1}{x^4}}## then compute it's second derivative and find it's max value, for the trapezoidal error formula.

Mark44 said:
Your integrand isn't right. What is ##\frac d{dx}\frac 1 {x^2}##? In your integrand above, it looks like you just squared ##\frac 1 {x^2}##, and forgot to find its derivative.
Yes sorry I realized that then fixed it.

Zack K said:
Yes sorry I realized that then fixed it.
It's still wrong. You need to find the derivative of 1/x^2, square it, and add 1. That will go inside the radical for your arc length integrand.

Mark44 said:
It's still wrong. You need to find the derivative of 1/x^2, square it, and add 1. That will go inside the radical for your arc length integrand.
Sigh... now I was thinking of integration.

It should be ##\sqrt{1+\frac{4}{x^6}}=\sqrt{\frac{x^6+4}{x^6}}##

Zack K said:
It should be ##\sqrt{1+\frac{4}{x^6}}=\sqrt{\frac{x^6+4}{x^6}}##
That's more like it.

Zack K said:
##E_T\leq\frac{(b-a)^3}{12n^2}[max |f^{(2)}(x)|]##
After you get the integral squared away, there are some tricks you can use to find the max of |f2(x)|. If the function (f2(x)) is increasing on an interval [a, b], its largest value will be at x = b. If the function is decreasing, it's largest value will be at the left endpoint of the interval, x = a.

Zack K
Mark44 said:
That's more like it.After you get the integral squared away, there are some tricks you can use to find the max of |f2(x)|. If the function (f2(x)) is increasing on an interval [a, b], its largest value will be at x = b. If the function is decreasing, it's largest value will be at the left endpoint of the interval, x = a.
Wow that actually makes so much sense. My biggest issue is once I got the second derivative, I got this large function which seemed annoying to evaluate with x=5 (given that this question could be on an exam). Thank you.

## 1. What is the trapezoidal rule for estimating arc length error?

The trapezoidal rule is a numerical method used to approximate the arc length of a curve. It involves dividing the curve into small trapezoids and finding the sum of their areas to estimate the total arc length.

## 2. How accurate is the trapezoidal rule for estimating arc length error?

The accuracy of the trapezoidal rule depends on the number of trapezoids used. The more trapezoids, the closer the estimate will be to the actual arc length. However, it is not as accurate as other numerical methods such as Simpson's rule.

## 3. Can the trapezoidal rule be used for any type of curve?

Yes, the trapezoidal rule can be used for any type of curve as long as it is continuous and has a known equation. It is commonly used for estimating arc length error in calculus and physics problems.

## 4. How do you calculate the error in the trapezoidal rule for estimating arc length?

The error in the trapezoidal rule can be calculated using the formula: E = -(b-a)^3/12n^2 * f''(c), where (b-a) is the interval of integration, n is the number of trapezoids, and f''(c) is the second derivative of the curve at some point c within the interval.

## 5. Are there any limitations to using the trapezoidal rule for estimating arc length error?

Yes, the trapezoidal rule can underestimate the arc length if the curve has sharp turns or high curvature. It also requires a known equation for the curve, which may not always be available. In these cases, other numerical methods may be more suitable.

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