Calculating Area of Polar Function: Spiral r = 5(e^.1θ)

• klarge
In summary, the problem is to find the area of the region in Quadrant I outside the second revolution and inside the third revolution of the polar graph r = 5(e^.1θ). The bounds for the second revolution are 2π < θ < 4π and for the third revolution are 4π < θ < 6π. To determine the bounds, it is helpful to know that one full revolution is 2π radians.
klarge
1. Problem: Use the spiral r = 5(e^.1θ). Find the area of the region in Quadrant I that is outside the second revolution of the spiral and inside the third revolution.

2. Homework Equations :

3. Attempt at solution:

My problem with finding the area of a polar graph is determining the bounds, so to get the right bounds for this graph do I set the equation equal to zero? I am really at a loss as to how to set up the bounds. Any hints would be helpful.

So... the second revolution is between which values of $\theta$ and the third is between which values of $\theta$? $\theta = 0$ is the start of the first.

2nd between -2pi and 2pi and the third between -4pi and 4pi?

One full revolution is $2\pi$ radians. If revolution 1 is for $0 < \theta < 2\pi$ and the second revolution starts at $\theta = 2\pi$ radians and spans $2\pi$ subsequent radians, the second revolution has what range of $\theta$?.

$2\pi < \theta < 4\pi$
and the third would be $4\pi < \theta < 6\pi$.

Correct.

So the bounds would be between $2\pi < \theta < 4\pi$?

1. What is the definition of "Area of Polar Function"?

The area of polar function is a mathematical concept used to calculate the total area enclosed by a polar curve on a polar coordinate system.

2. How is the area of polar function different from calculating the area of a regular function?

The area of polar function involves integrating the polar equation over a certain interval, while calculating the area of a regular function involves finding the area under a curve on a Cartesian coordinate system. Additionally, polar coordinates use radial distance and angle, while Cartesian coordinates use x and y coordinates.

3. What is the formula for finding the area of polar function?

The formula for finding the area of polar function is A = ½∫ab r2 dθ, where r is the polar equation and θ is the angle of rotation.

4. Can the area of polar function be negative?

No, the area of polar function cannot be negative. It represents the total enclosed area, which is always a positive value.

5. What are some real-world applications of the area of polar function?

The area of polar function is used in various fields such as physics, engineering, and astronomy to calculate the areas of objects with circular or spiral shapes. It is also used in navigation and mapping to determine the area of land masses or bodies of water.

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