Understanding Polar Curves and Area Calculation

The video tutorial explains how to find the area inside the loop of a polar curve defined by the equation r = 3 - 6 sin(θ). It covers two methods to determine the interval of integration: using a graphing calculator and solving the equation by setting r to zero. The tutorial then demonstrates how to evaluate the integral to find the area, including a substitution for sine squared theta and a u-substitution for cosine two theta. Finally, the video verifies the result using a graphing calculator, ensuring the calculated area is accurate.

What is the equation of the polar curve whose inner loop area we are trying to find?

• r = 3 - 6 sin(θ)

• r = 3 + 6 sin(θ)

• r = 6 + 3 sin(θ)

• r = 6 - 3 sin(θ)

Which formula is used to find the area bounded by a polar curve?

• Area = integral of r dθ

• Area = integral of r^2 dθ

• Area = 1/2 integral of r^2 dθ

• Area = 1/2 integral of r dθ

What is the interval of integration for tracing the inner loop using the graphing calculator?

• 0 to 180 degrees

• 30 to 150 degrees

• 60 to 120 degrees

• 90 to 180 degrees

How can the interval of integration be determined analytically?

• By setting r = 1

• By setting r = 0

• By setting θ = π

• By setting θ = 0

What substitution is used for simplifying the sine squared term in the integral?

• 1/2 (1 - sin(2θ))

• 1/2 (1 + cos(2θ))

• 1/2 (1 - cos(2θ))

• 1/2 (1 + sin(2θ))

What is the antiderivative of 27/2 with respect to θ?

• 13.5 θ

• 27 θ

• 27/2 θ

• 54 θ

What is the approximate area of the inner loop calculated in the video?

• 5.8917 square units

• 3.8917 square units

• 2.8917 square units

• 4.8917 square units

Why is it important to switch the calculator back to radian mode before evaluating the integral?

• To avoid errors in trigonometric calculations

• Because the integral is defined in radians

• Because the integral is defined in degrees

• To ensure the calculator functions properly

What is the purpose of using a graphing calculator in this problem?

• To find the derivative of the function

• To verify the calculated area

• To solve the equation for θ

• To graph the curve in Cartesian coordinates

What is the significance of the interval from π/6 to 5π/6 in this problem?

• It represents the maximum area

• It represents the inner loop

• It represents the outer loop

• It represents the entire curve