Use the arc length formula to calculate the exact length of the curve. (Use a calculator to evaluate the integral.)
The correct answer and explanation is:
Sure! To help you accurately, I need a bit more detail:
- What is the function y=f(x)y = f(x) or parametric form of the curve?
- What is the interval for xx (or parameter) over which you want to find the arc length?
Meanwhile, here’s a general explanation of how to calculate the exact length of a curve using the arc length formula:
Arc Length Formula for a Curve y=f(x)y = f(x)
The arc length LL of a smooth curve y=f(x)y = f(x) from x=ax = a to x=bx = b is given by: L=∫ab1+(dydx)2 dxL = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
Steps to Calculate Arc Length:
- Find the derivative: Compute dydx\frac{dy}{dx} of the function y=f(x)y = f(x).
- Square the derivative: Calculate (dydx)2\left(\frac{dy}{dx}\right)^2.
- Add 1: Form the expression inside the square root: 1+(dydx)21 + \left(\frac{dy}{dx}\right)^2.
- Set up the integral: Integrate the square root over the interval a≤x≤ba \leq x \leq b.
- Evaluate the integral: Use a calculator or numerical methods to find the approximate value of the integral.
Example:
Suppose y=x2y = x^2 over the interval [0,1][0, 1].
- dydx=2x\frac{dy}{dx} = 2x.
- (dydx)2=4×2\left(\frac{dy}{dx}\right)^2 = 4x^2.
- The integral is
L=∫011+4×2 dxL = \int_0^1 \sqrt{1 + 4x^2} \, dx
- This integral doesn’t have a simple antiderivative, so we approximate it numerically using a calculator.
- Using numerical integration (like Simpson’s rule or a calculator’s integral function), the length is approximately 1.4789.
Summary:
- The arc length formula transforms the geometric problem of finding curve length into a calculus integral.
- The main challenge is evaluating the integral ∫ab1+(dy/dx)2dx\int_a^b \sqrt{1 + (dy/dx)^2} dx, often requiring numerical methods.
- Calculators and software tools help find accurate numerical results when no elementary antiderivative exists.
- This method applies to any differentiable function, and the same logic extends to parametric and polar curves with modified formulas.
If you provide the exact function and interval, I can do the full calculation for you with the exact and numerical result!