How do you find the arc length of $y=x^2$ from $x=0$ to $x=3$?
1. Find $f'(x) = 2x$. 2. Apply the arc length formula: $S=\int_0^3 \sqrt{1+(2x)^2} dx$. 3. Evaluate the integral (approximately 3.823).
Steps to calculate distance traveled along a curve?
1. Define the function $f(x)$. 2. Find the derivative $f'(x)$. 3. Apply the arc length formula $S = \int_a^b \sqrt{1 + [f'(x)]^2} dx$. 4. Evaluate the integral.
What is arc length?
The distance along a curve.
Define a smooth, planar curve.
A curve in a two-dimensional plane that has a continuous derivative.
What is distance traveled?
The total length of the path covered by an object in motion.
Define the integrand in the arc length formula.
The term inside the integral, $\sqrt{1+[f'(x)]^2}$, which calculates the length of an infinitesimally small section of the curve.
What is the arc length formula for $y=f(x)$ from $x=a$ to $x=b$?
$S=\int_a^b \sqrt{1+[f'(x)]^2} dx$
What does $f'(x)$ represent in the arc length formula?
The derivative of the function $f(x)$ with respect to $x$.
