Arc Length in Polar Coordinates

The video tutorial explains how to derive the formula for the arc length of a curve defined in polar coordinates. It starts by introducing the concept of arc length and infinitesimal segments, then relates polar coordinates to rectangular coordinates. The tutorial uses the product rule to derive expressions for DX and DY, calculates DS, and simplifies the expression to arrive at the final arc length formula in polar form.

What is the main goal of the video tutorial?

• To find the arc length of a curve in polar coordinates.

• To discuss the history of calculus.

• To explain the concept of polar coordinates.

• To derive the formula for the area of a circle.

What is the infinitesimal element of arc length called?

• DR

• DT

• DS

• DX

How is the total arc length obtained from the infinitesimal elements?

• By multiplying them together.

• By integrating them.

• By adding them directly.

• By subtracting them.

What is the relationship between X and r in polar coordinates?

• X = r * sin(theta)

• X = r / cos(theta)

• X = r / sin(theta)

• X = r * cos(theta)

Which rule is applied to find DX and DY?

• Chain Rule

• Quotient Rule

• Product Rule

• Power Rule

What is the expression for DS squared derived from?

• The Law of Sines

• The Pythagorean Theorem

• The Fundamental Theorem of Calculus

• The Law of Cosines

What happens to the middle terms when simplifying DX squared and DY squared?

• They double.

• They cancel out.

• They become zero.

• They remain unchanged.

What is the final step in deriving the arc length formula?

• Taking the derivative

• Taking the square root

• Multiplying by theta

• Dividing by theta

What is the integral used for in the final formula?

• To find the area under the curve

• To calculate the total arc length

• To determine the slope of the curve

• To find the maximum value of r

What is the range of integration for the arc length formula?

• From 0 to 2π

• From alpha to beta

• From -∞ to ∞

• From 0 to 1