Arc Length and Derivatives in Polar Coordinates

The video tutorial covers the calculation of arc lengths for various polar curves using calculus. It begins with the formula for arc length and applies it to different functions, including r = 6 sin(theta), r = 1 + sin(theta), r = 5, and r = 2theta. The tutorial includes graphical representations and verifies results using geometric properties like circle circumference. Each example demonstrates the integration process and highlights key trigonometric identities.

What is the formula for finding the arc length of a polar curve?

• Integral from alpha to beta of r dθ

• Integral from alpha to beta of (r^2 + (dr/dθ)^2) dθ

• Integral from alpha to beta of √(r^2 + (dr/dθ)^2) dθ

• Integral from alpha to beta of (r^2 - (dr/dθ)^2) dθ

What is the derivative of 6 sin(θ) with respect to θ?

• sin(θ)

• 6 cos(θ)

• 6 sin(θ)

• cos(θ)

What identity is used to simplify the expression 36 sin²(θ) + 36 cos²(θ)?

• Euler's identity

• Trigonometric identity

• Pythagorean identity

• Binomial identity

What is the arc length of the curve r = 6 sin(θ) from 0 to π/2?

• 2π

• 3π

• π

• 6π

How is the curve r = 6 sin(θ) geometrically interpreted?

• An ellipse

• A parabola

• A line

• A circle

What is the arc length of the curve r = 1 + sin(θ) from 0 to 2π?

• 4

• 8

• 10

• 6

What is the derivative of 1 + sin(θ) with respect to θ?

• 1 + sin(θ)

• 1 + cos(θ)

• sin(θ)

• cos(θ)

What is the arc length of the curve r = 5 from 0 to 2π?

• 15π

• 10π

• 5π

• 20π

What is the derivative of 2θ with respect to θ?

• 0

• 2θ

• θ

• 2

What is the approximate arc length of the curve r = 2θ from 0 to π/2?

• 4.158

• 5.158

• 3.158

• 6.158