How To Find Arc Length

How to Find Arc Length in Standard American English

The arc length of a curve is the distance along the curve between two points. It can be calculated using the following formula:

s = ∫[a,b] √(1 + (dy/dx)^2) dx

where:

• s is the arc length
• a and b are the lower and upper bounds of the integral
• y is the function that defines the curve
• dy/dx is the derivative of y

To evaluate this integral, you can use a variety of techniques, including integration by substitution, integration by parts, and numerical integration.

Integration by substitution

If the integrand can be expressed in terms of a single variable, you can use integration by substitution to evaluate the integral. For example, if the integrand is √(1 + (dy/dx)^2), you can substitute u = dy/dx. Then, du/dx = d^2y/dx^2, and the integral becomes:

s = ∫[a,b] √(1 + u^2) du

This integral can be evaluated using the following formula:

∫[a,b] √(1 + u^2) du = sinh^-1(u) + C

where C is a constant of integration.

Integration by parts

If the integrand cannot be expressed in terms of a single variable, you can use integration by parts to evaluate the integral. For example, if the integrand is x√(1 + y^2), you can use the following formula:

∫[a,b] uv dx = uv - ∫[a,b] v du

where u and v are functions of x.

In this case, you can choose u = x and v = √(1 + y^2). Then, du/dx = 1 and dv/dx = y/√(1 + y^2). The integral becomes:

s = ∫[a,b] x√(1 + y^2) dx

= x√(1 + y^2) - ∫[a,b] √(1 + y^2) dx

The second integral can be evaluated using the formula for the integral of √(1 + u^2).

Numerical integration

If the integrand cannot be evaluated analytically, you can use numerical integration to approximate the value of the integral. There are a variety of numerical integration methods, including the trapezoidal rule, the Simpson’s rule, and the Monte Carlo method.

The trapezoidal rule approximates the integral as the sum of the areas of trapezoids that are formed by dividing the interval [a,b] into n subintervals. The formula for the trapezoidal rule is:

s ≈ (b - a)/2n * [f(a) + 2f(a + h) + 2f(a + 2h) + ... + 2f(a + (n-1)h) + f(b)]

where h = (b – a)/n.

The Simpson’s rule approximates the integral as the sum of the areas of parabolas that are formed by dividing the interval [a,b] into n subintervals. The formula for the Simpson’s rule is:

s ≈ (b - a)/6n * [f(a) + 4f(a + h) + 2f(a + 2h) + 4f(a + 3h) + ... + 2f(a + (n-1)h) + 4f(a + nh) + f(b)]

The Monte Carlo method approximates the integral by randomly generating points in the region under the curve and then taking the average of the values of the function at those points. The formula for the Monte Carlo method is:

s ≈ (b - a)*mean(f(x))

where f(x) is the function that defines the curve.

Example

Find the arc length of the curve y = x^2 from x = 0 to x = 1.

Solution

The arc length of the curve is given by the following integral:

s = ∫[0,1] √(1 + (dy/dx)^2) dx

= ∫[0,1] √(1 + 4x^2) dx

This integral can be evaluated using the following formula:

∫[0,1] √(1 + 4x^2) dx = (1/4)sinh^-1(2x) + C

where C is a constant of integration.

Evaluating the integral from x = 0 to x = 1, we get:

s = (1/4)sinh^-1(2) - (1/4)sinh^-1(0)

= (1/4)ln(2 + √3)

Therefore, the arc length of the curve y = x^2 from x = 0 to x = 1 is (1/4)ln(2 + √3).

FAQ

Q: What is the difference between arc length and path length?

A: Arc length is the distance along a curve, while path length is the total distance traveled along a path. The arc length of a curve is always less than or equal to the path length.

Q: How do I find the arc length of a curve that is not smooth?

A: To find the arc length of a curve that is not smooth, you can use the following formula:

s = lim[n→∞] ∑[i=1,n] √((x_i - x_(i-1))^2 + (y_i - y_(i-1))^2)

where (x_i, y_i) are the coordinates of the points on the curve that divide the curve into n subintervals.

Q: How do I find the arc length of a curve in polar coordinates?

A: To find the arc length of a curve in polar coordinates, you can use the following formula:

s = ∫[a,b] √(r^2 + (dr/dθ)^2) dθ

where r is the radius of the curve and θ is the angle that the curve makes with the positive x-axis.