# Compute the length of the cardioid below the x-axis?

• IntegrateMe
In summary, the conversation discusses finding the limits of integration for a given formula. The answer key suggests using 0 to π, but the speaker suggests using π to 2π instead. They also mention considering the graph to determine the direction when the formula crosses the x-axis.
IntegrateMe
$$r = 2sin(\theta)-2$$

First we find x(θ), y(θ)

$$x(\theta) = rcos(\theta)$$
$$y(\theta) = rsin(\theta)$$

Then we find x'(θ) and y'(θ) to use the formula:

$$L = \int_\alpha^β \sqrt{x'(\theta)^2 + y'(\theta)^2} d\theta$$

My problem is that I don't know how to get the limits of integration. The answer key says that they are from 0 to π, but I would have guessed π to 2π, since that represents everything below the x-axis? Any help would be appreciated.

Thanks, guys!

Why guess? It's going to cross the x-axis when y = 0. What values of ##\theta## do that? Plot the graph for additional information on what direction it is going when it crosses the x axis.

## 1. What is a cardioid?

A cardioid is a geometric shape that resembles a heart or a circle with a cusp. It is a type of curve known as a limaçon and is commonly used in mathematics and physics.

## 2. How do you compute the length of a cardioid?

To compute the length of a cardioid below the x-axis, you can use the formula L = 8a, where a is the distance from the origin to the cusp of the cardioid. This formula is derived from the polar equation r = a(1 + cosθ).

## 3. What is the significance of computing the length of a cardioid below the x-axis?

Computing the length of a cardioid below the x-axis can be useful in various applications, such as determining the area enclosed by the cardioid or calculating the arc length of a curve. It is also a common exercise in mathematics and can help improve problem-solving skills.

## 4. Can the length of a cardioid below the x-axis be negative?

No, the length of a cardioid below the x-axis cannot be negative. The length of a curve is always a positive value, as it represents a physical distance. The formula for computing the length of a cardioid below the x-axis ensures that the result is always positive.

## 5. Are there any real-world applications for the length of a cardioid below the x-axis?

Yes, there are several real-world applications for the length of a cardioid below the x-axis. For example, cardioids are often used to model the trajectory of objects in motion, such as the path of a thrown ball or the motion of a pendulum. In these cases, computing the length of a cardioid can help determine the distance traveled by the object.

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