The two values for $\theta$ would be 0 and $\pi$.

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In summary, a lemniscate is a mathematical curve that resembles a figure-eight shape. The area enclosed by a lemniscate is calculated using the formula A = 2πa², and it represents the maximum area that can be enclosed by a curve with a given length. The area cannot be negative and has applications in physics and engineering. Real-life examples of lemniscates include planetary orbits, a twirling lasso, and certain flower petals.
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veronica1999
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Question: What is the area enclosed by the lemniscate r2 = -25cos(Ѳ)?
The answer is 25.

I can't seem to set up the integral for this question correctly. I know that the area enclosed by a polar curve is obtained by the integral(r2/2) dѲ, but I can't determine what the interval of integration would be.
Help please!
 
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Let's take a look at a plot of the curve:

View attachment 2452

We want to begin and end at the origin...what two values for $\theta$ on $[0,2\pi)$ will put you at the origin?
 

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FAQ: The two values for $\theta$ would be 0 and $\pi$.

What is a lemniscate?

A lemniscate is a mathematical curve that resembles a figure-eight shape. It is also known as the infinity symbol (∞).

How is the area enclosed by a lemniscate calculated?

The area enclosed by a lemniscate is calculated using the formula A = 2πa², where a is the distance from the center of the lemniscate to any point on the curve.

What is the significance of the area enclosed by a lemniscate?

The area enclosed by a lemniscate is significant because it represents the maximum area that can be enclosed by a curve with a given length. It also has applications in physics and engineering, such as in the study of planetary orbits.

Can the area enclosed by a lemniscate be negative?

No, the area enclosed by a lemniscate cannot be negative. It is always a positive value, since it represents a physical area.

Is there a real-life example of a lemniscate?

Yes, there are many real-life examples of lemniscates, such as the orbits of planets and the shape of a lasso when it is twirled. The shape of a lemniscate can also be seen in the petals of some flowers, such as the anemone and the hellebore.

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