Arc Length of a Spiral Polar Curve

The video tutorial explains how to find the arc length of a spiral polar curve defined by R = 2 * e^(3θ) from 0 to π/3 radians. It covers the derivation of the arc length formula, the calculation of derivatives using the chain rule, and the simplification of the integrand. The tutorial then demonstrates the integration process, including u-substitution, to find the exact arc length. Finally, it provides a decimal approximation of the arc length and concludes with a summary of the key points.

What is the polar curve equation given in the problem?

• R = 2 * e^(3θ)

• R = 3 * e^(2θ)

• R = 2 * e^(θ/3)

• R = 3 * e^(θ/2)

What is the interval of integration for the arc length calculation?

• 0 to 2π radians

• 0 to π/2 radians

• 0 to π radians

• 0 to π/3 radians

Which rule is applied to find the derivative of R with respect to θ?

• Chain Rule

• Quotient Rule

• Power Rule

• Product Rule

What is the derivative of R with respect to θ?

• 6 * e^(3θ)

• 3 * e^(2θ)

• 2 * e^(3θ)

• e^(3θ)

What is the expression under the square root after simplification?

• 30 * e^(3θ)

• 40 * e^(6θ)

• 20 * e^(3θ)

• 10 * e^(6θ)

How is the square root of e^(6θ) expressed using rational exponents?

• e^(6θ/12)

• e^(3θ)

• e^(6θ/2)

• e^(6θ/3)

What constant is factored out before integration?

• √30

• √40

• √20

• √10

What substitution is used during integration?

• U = θ/3

• U = θ/6

• U = 6θ

• U = 3θ

What is the exact arc length of the curve from 0 to π/3 radians?

• 2√10/3 * (e^π - 1)

• 2√10/3 * (e^π + 1)

• 2√10/3 * (e^π/2 - 1)

• 2√10/3 * (e^π/2 + 1)

What is the approximate arc length of the curve?

• 43.67

• 45.67

• 46.67

• 44.67