Area of one loop of the rose r=cos(3theta)

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In summary, the question asks how to use double integrals to find the area of one loop of the rose r=cos(3theta). The values of -pi/6 and pi/6 for theta are chosen because they result in r=0 which is the origin, thus creating a closed loop. Since cos(3theta) is not 0 between -pi/6 and pi/6, this encloses only one loop. The conversation ends with the understanding and gratitude of the asker.
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Ortix
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Homework Statement


Use double integrals to find the Area of one loop of the rose r=cos(3theta)

I know how to solve this, the only question I have is why theta is between -pi/6 and pi/6. I don't understand where those two values come from.
 
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  • #2
With theta equal to -pi/6, 3theta= -pi/2 and r= cos(3theta)= cos(-pi/2)= 0. Similarly, if theta is pi/6, 3theta= pi/2 and r= cos(3theta)= cos(pi/2)= 0. The only point with r= 0 is the origin, no matter what theta is. That means that starting at -pi/6 and going to pi/6 you have closed a loop. starting at the origin and coming back to it. Since cosine in not 0 at any point between -pi/2 and pi/2, cos(3theta) is not 0 at any point between -pi/6 and pi/6 so that encloses only one loop which is what you want.,
 
  • #3
finally! Now i understand it! Thank you so much! :)
 

FAQ: Area of one loop of the rose r=cos(3theta)

What is the equation for calculating the area of one loop of the rose r=cos(3theta)?

The equation for calculating the area of one loop of the rose r=cos(3theta) is A = (3pi/8)r^2.

What does the variable "r" represent in the equation for the area of one loop of the rose r=cos(3theta)?

In this equation, "r" represents the radius of the loop of the rose. It is equal to the cosine of three times theta.

How do you graph the rose r=cos(3theta) to visualize the area of one loop?

To graph the rose r=cos(3theta), plot points with the coordinates (rcos(3theta), rsin(3theta)) for different values of theta. The resulting graph will have a petal-like shape and the area of one loop can be visualized within the graph.

What is the significance of the number "3" in the equation for the area of one loop of the rose r=cos(3theta)?

The number "3" represents the number of loops in the rose. In this equation, there will be three loops in the rose, resulting in a total area of (3pi/8)r^2.

How is the area of one loop of the rose r=cos(3theta) related to the total area of the rose?

The area of one loop of the rose r=cos(3theta) is equal to one-third of the total area of the rose. This is because the rose has three loops, and the area of one loop is (3pi/8)r^2, while the total area is (9pi/8)r^2.

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