Revise later
SpaceTo flip
If confident
All Flashcards
Explain how the Pythagorean Theorem relates to the arc length formula.
The arc length formula uses the Pythagorean theorem ($c=\sqrt{a^2+b^2}$) to find the length of small segments on the curve, where 1 represents the square of the length along the x-axis and $[f'(x)]^2$ represents the square of the length along the y-axis.
How does arc length relate to distance traveled?
Arc length provides a way to calculate the distance traveled along a curve, even when the curve is complex and continuously changing.
Why is arc length important in calculus?
It allows us to calculate the distance traveled along a curve, which is crucial in various applications in mathematics and physics.
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 $x$.
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.