Arc Length of Polar Curves

The video tutorial explains how to find the arc length of a polar curve defined by R = 5 + 5 cos(Theta). It introduces the arc length formula and demonstrates its application using symmetry to simplify calculations. The tutorial walks through the process of simplifying the integral and performing the final calculation, resulting in an arc length of 40 units.

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

• L = integral of R^2 dTheta

• L = integral of sqrt(R^2 + (dR/dTheta)^2) dTheta

• L = integral of (dR/dTheta)^2 dTheta

• L = integral of R dTheta

Why can we integrate from 0 to Pi and then double the result when finding the arc length of the given polar curve?

• Because the curve is a straight line

• Because the curve is symmetric about the y-axis

• Because the curve is symmetric about the polar axis

• Because the curve is a full circle

What is the derivative of R with respect to Theta for the curve R = 5 + 5 cos(Theta)?

• 5 cos(Theta)

• -5 cos(Theta)

• -5 sin(Theta)

• 5 sin(Theta)

What is the result of squaring the expression 5 + 5 cos(Theta)?

• 50 + 25 cos(Theta)

• 25 + 50 cos(Theta) + 25 cos^2(Theta)

• 50 + 50 cos(Theta)

• 25 + 25 cos(Theta)

Which trigonometric identity is used to simplify the integrand in the arc length calculation?

• sin^2(Theta) + cos^2(Theta) = 1

• 1 + cos(2Theta) = 2cos^2(Theta)

• tan^2(Theta) + 1 = sec^2(Theta)

• 1 - cos(2Theta) = 2sin^2(Theta)

What substitution is used to simplify the integration of the square root of 1 + cos(Theta)?

• U = 2Theta

• U = Theta

• U = cos(Theta)

• U = Theta/2

What is the final arc length of the polar curve R = 5 + 5 cos(Theta)?

• 50 units

• 20 units

• 30 units

• 40 units

What is the value of sin(Pi/2) used in the final calculation of the arc length?

• -1

• 1

• 0

• 0.5

What is the purpose of multiplying the integral by 2 in the arc length calculation?

• To simplify the integration process

• To correct a calculation error

• To account for the full revolution of the curve

• To adjust for the units of measurement

Which of the following is a correct step in simplifying the integrand for the arc length?

• Multiplying by 2

• Using the identity sin^2(Theta) + cos^2(Theta) = 1

• Factoring out 100

• Dividing by 2