How does arc length relate to the Pythagorean Theorem?

Arc length is approximated by summing the hypotenuses of small right triangles with sides dx and dy.

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How does arc length relate to the Pythagorean Theorem?

Arc length is approximated by summing the hypotenuses of small right triangles with sides dx and dy.

Why do we need a different arc length formula for parametric curves?

Because both x and y coordinates are changing with respect to a parameter t, so we must account for both rates of change.

Arc length formula for parametric curves?

$S=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt$

Arc length formula for Cartesian curves?

$S=\int_a^b \sqrt{1+[f'(x)]^2} dx$

Define arc length.

The distance between two points along a curve.

What is a parametric curve?

A curve where the x and y coordinates are defined by functions of a parameter, usually denoted as t.

Define a smooth planar curve.

A curve in a plane that has a continuous first derivative.

All Flashcards

How does arc length relate to the Pythagorean Theorem?

Arc length is approximated by summing the hypotenuses of small right triangles with sides dx and dy.

Why do we need a different arc length formula for parametric curves?

Because both x and y coordinates are changing with respect to a parameter t, so we must account for both rates of change.

Arc length formula for parametric curves?

$S=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt$

Arc length formula for Cartesian curves?

$S=\int_a^b \sqrt{1+[f'(x)]^2} dx$

Define arc length.

The distance between two points along a curve.

What is a parametric curve?

A curve where the x and y coordinates are defined by functions of a parameter, usually denoted as t.

Define a smooth planar curve.

A curve in a plane that has a continuous first derivative.