Integrals and Polar Curves

This video tutorial explains how to calculate the area bounded by two polar curves. It begins with a review of the formula for finding the area between two polar curves and then provides a detailed example. The example demonstrates how to set up the problem, verify integration limits using a graphing calculator, and evaluate the definite integrals. The video emphasizes the importance of correctly identifying the outer and inner functions when integrating with respect to theta. The tutorial concludes with a summary of the steps and a reminder to be cautious when setting up such problems.

What is the formula for the area bounded by two polar curves?

• One half times the integral of the sum of the squares of the functions

• The integral of the sum of the functions

• One half times the integral of the difference of the squares of the functions

• The integral of the product of the functions

In the example problem, what is the outer function when theta is between 0 and pi/6?

• r = 5

• r = 2

• r = 3 cos(theta)

• r = 4 sin(theta)

Why is it necessary to use two definite integrals in the example problem?

• Because the area is bounded by a single curve

• Because the curves intersect at multiple points

• Because the area is bounded by two different curves in different intervals

• Because the area is unbounded

What mode should the graphing calculator be set to for this problem?

• Radian and Cartesian mode

• Degree and Cartesian mode

• Degree and Polar mode

• Radian and Polar mode

What is the purpose of using the graphing calculator in this example?

• To find the exact area of the region

• To verify the limits of integration and visualize the problem

• To solve the integrals directly

• To check the correctness of the formula

Which formula is applied to simplify the integral of sine squared theta?

• Product-to-sum formula

• Sum-to-product formula

• Double angle formula

• Power-reducing formula

What is the result of applying the power-reducing formula to 16 sine squared theta?

• 16 + 16 cosine 2 theta

• 8 + 8 cosine 2 theta

• 16 - 16 cosine 2 theta

• 8 - 8 cosine 2 theta

What additional step is required when performing U-substitution in this problem?

• Multiplying by the derivative of the inner function

• Adding an extra factor of one half

• Subtracting a constant

• Adding an extra factor of two

What is the final expression for the area of the shaded region?

• 10 pi thirds plus 4 square root 3

• 8 pi thirds minus 2 square root 3

• 4 pi thirds plus 2 square root 3

• 6 pi thirds minus 3 square root 3

What is emphasized as important when setting up the integrals for polar curves?

• Ensuring the curves are linear

• Checking the area is bounded with respect to theta

• Verifying the curves intersect at the origin

• Using Cartesian coordinates