Polar Coordinates Area and Integration

Polar Coordinates Area and Integration

Assessment

Interactive Video

Mathematics

11th - 12th Grade

Practice Problem

Hard

Created by

Thomas White

FREE Resource

This video tutorial explains how to find areas using polar coordinates, focusing on graphing and determining integration limits. It provides a step-by-step example of finding the area enclosed by one loop of a polar curve, R = sin(4θ). The tutorial emphasizes the importance of graphing to determine integration limits and demonstrates the integration process using trigonometric identities. The video concludes with a brief mention of more complex examples involving areas between two polar curves.

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7 questions

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1.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the main challenge in finding areas in polar coordinates according to the video?

Calculating the radius

Finding a decent graph and limits of integration

Understanding the concept of angles

Using trigonometric identities

2.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

In polar coordinates, what does 'R' represent?

The area

The angle

The distance from the origin

The circumference

3.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the key difference in limits of integration between polar and rectangular coordinates?

Rectangular limits are based on angles

Rectangular limits are based on radii

Polar limits are based on angles

Polar limits are based on distances

4.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the function used in the example problem to find the area enclosed by one loop?

R = cos(θ)

R = tan(θ)

R = sin(4θ)

R = cos(2θ)

5.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

How do you determine the limits of integration for the example problem?

By finding when the angle is maximum

By finding when the angle is zero

By finding when the radius is zero

By finding when the radius is maximum

6.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Which trigonometric identity is used in the integration process?

cos^2(θ) = 1 - sin^2(θ)

tan^2(θ) = sec^2(θ) - 1

sin^2(θ) = 1/2(1 - cos(2θ))

cos^2(θ) = 1/2(1 + cos(2θ))

7.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the final calculated area for the example problem?

π/2

π/16

π/4

π/8

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