estimate the arc length of the curve y=sinx for 0<x<pi
Answer
Estimating the Arc Length of y = sin(x) from 0 to π
To find the arc length of a curve defined by a function y = f(x) over an interval, we use the standard formula from calculus. The curve y = sin(x) is smooth and continuous between 0 and π, making it ideal for using the arc length formula directly.
📘 Arc Length Formula
In this case, f(x) = sin(x). First, we compute the derivative:
Now plug this into the arc length formula:
🔍 Why Can’t This Be Integrated Exactly?
The integral ∫ √(1 + cos²(x)) dx does not have an elementary antiderivative, so we estimate it numerically, using methods like:
- Trapezoidal Rule
- Simpson’s Rule
- Numerical Integration Tools
💡 Numerical Estimation
Using numerical integration (such as Simpson’s Rule or a calculator), we estimate:
So, the arc length of the sine curve from x = 0 to x = π is approximately 3.82 units.
🎯 Final Result
The estimated arc length of the curve y = sin(x) from 0 to π is: