Area of a Cardioid - Littlehime's Question @ Yahoo Answers

Therefore, the area inside the cardioid is 24π square units.In summary, the given problem asks to find the area inside a cardioid with polar equation r=4+4cos(θ) for 0≤θ≤2π. Using the formula A=½∫r^2dθ, we can solve for the area by expanding the integrand and applying the FTOC. The final result is 24π square units.
  • #1
MarkFL
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Here is the question:

Help finding the area of this cardioid 10 points best?


Find the area inside the cardioid r=4+4cos(θ) for 0 less than/equal to θ less than/equal to 2pi

I have posted a link there to this thread so the OP can see my work.
 
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  • #2
Hello Littlehime,

The area in polar coordinates is given by:

\(\displaystyle A=\frac{1}{2}\int_{\alpha}^{\beta}r^2\,d\theta\)

For the given problem, we are told:

\(\displaystyle \alpha=0,\,\beta=2\pi,\,r=4\left(1+\cos(\theta) \right)\)

And so we have:

\(\displaystyle A=8\int_{0}^{2\pi} \left(1+\cos(\theta) \right)^2\,d\theta\)

Expanding the integrand, we may write:

\(\displaystyle A=8\int_{0}^{2\pi} 1+2\cos(\theta)+\cos^2(\theta)\,d\theta\)

Applying a double-angle identity for cosine, we obtain:

\(\displaystyle A=4\int_{0}^{2\pi} 4\cos(\theta)+\cos(2\theta)+3\,d\theta\)

Applying the FTOC, we get:

\(\displaystyle A=4\left[4\sin(\theta)+\frac{1}{2}\sin(2\theta)+3\theta \right]_0^{2\pi}=4\left(3(2\pi) \right)=24\pi\)
 

FAQ: Area of a Cardioid - Littlehime's Question @ Yahoo Answers

1. What is a cardioid?

A cardioid is a mathematical shape that resembles a heart or a four-leaf clover. It is a type of curve known as an epicycloid, and is created by tracing a fixed point on a circle as it rolls around another fixed circle.

2. What is the formula for finding the area of a cardioid?

The formula for finding the area of a cardioid is (3/4)πr², where r is the radius of the circle that forms the cardioid. This formula is derived from the general formula for finding the area of an epicycloid, which is (n-1/n)πr², with n being the number of cusps or loops in the shape.

3. How is the area of a cardioid related to its circumference?

The area of a cardioid is exactly half of its circumference multiplied by the radius. This relationship can be expressed mathematically as A = (1/2)C(r), where A is the area, C is the circumference, and r is the radius.

4. Are there any real-life applications of the cardioid?

Yes, the cardioid has several real-life applications. It is commonly used in the design of mechanical cams and gears, as well as in the design of audio equipment such as microphones and speakers. The shape is also found in nature, such as in the petals of certain flowers and the paths of some insects' flight patterns.

5. How can the area of a cardioid be visualized?

The area of a cardioid can be visualized by plotting points on a graph and connecting them to form the shape. Alternatively, a string can be attached to a pencil and a fixed point, and then traced as the pencil moves around a circle, creating a cardioid shape. There are also various online tools and simulations that can help visualize the area of a cardioid.

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