Understanding Area Calculation Using Polar Coordinates

This video tutorial explains how to calculate the area of the inner loop of a polar curve using integration. It covers setting up the integral, determining limits of integration, simplifying the integral with trigonometric identities, and evaluating the integral to find the exact area. The tutorial also provides a visual representation of the polar curve and discusses the orientation and plotting of points in polar coordinates.

What is the polar equation used to find the area of the inner loop in this lesson?

• r = 2 + cos(θ)

• r = 1 + 2cos(θ)

• r = 1 + 2sin(θ)

• r = 2 + 2cos(θ)

What is the significance of the interval from 2/3π to 4/3π radians in this lesson?

• It traces half of the inner loop.

• It traces the entire inner loop.

• It traces half of the outer loop.

• It traces the entire outer loop.

Why do we double the area when integrating from 2/3π to π radians?

• To account for the entire outer loop.

• To account for the entire inner loop.

• To account for half of the inner loop.

• To account for half of the outer loop.

What is the purpose of using a TI-84 calculator in this lesson?

• To graph the polar equation.

• To verify the limits of integration.

• To calculate the area directly.

• To convert radians to degrees.

Which trigonometric identity is used to simplify the integrand in this lesson?

• Power reducing formula for sine

• Power reducing formula for cosine

• Sine double angle identity

• Cosine double angle identity

What is the antiderivative of 3 with respect to θ?

• 3tan(θ)

• 3cos(θ)

• 3sin(θ)

• 3θ

What substitution is used to integrate 2cos(2θ)?

• u = sin(θ)

• u = cos(θ)

• u = 2θ

• u = θ

What is the exact area of the inner loop as derived in this lesson?

• π - 3√3/2

• 2π - 3√3

• π + 3√3/2

• 2π + 3√3

How is the expression for the area simplified to have a common denominator?

• By dividing by 3

• By multiplying by 2

• By dividing by 2

• By multiplying by 3

What alternative expression is given for the exact area of the inner loop?

• 2π/1 - 3√3

• 2π - 3√3/2

• π/2 - 3√3

• π/1 - 3√3/2