Polar Coordinates Homework: Integral w/ Image & Equations

In summary, polar coordinates are a system of describing points in a two-dimensional plane using a distance from the origin and an angle from a fixed reference line. To graph polar coordinates, the distance from the origin is plotted on the y-axis and the angle from the reference line is plotted on the x-axis. An integral in polar coordinates is a mathematical operation used to calculate the area under a curve in a polar graph. An example of solving an integral in polar coordinates is finding the area under a circle with a radius of 2, resulting in an answer of 16π. Polar coordinates are commonly used in fields such as physics, engineering, and navigation to describe circular or rotational motion, and in mapping and location-based applications to determine distance and direction
  • #1
dwn
165
2

Homework Statement



Image attached

Homework Equations



r2=x2+y2

The Attempt at a Solution



∫∫ re-r^2 drdΘ

I'm not sure how to establish the boundaries. This is an online class so if you can offer any additional tips for evaluating types of integrals of this sort, that would be great. Thank you.
 

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  • #2
dwn said:
I'm not sure how to establish the boundaries.
Try to graph the region.
If you want to approach it purely algebraically, the bounds are 0 < x < 1, 0< y < √(1-x2). Substitute the polar forms for x and y.
 

FAQ: Polar Coordinates Homework: Integral w/ Image & Equations

1. What are polar coordinates?

Polar coordinates are a system of describing points in a two-dimensional plane using a distance from the origin (known as the radius) and an angle from a fixed reference line (known as the polar axis).

2. How do you graph polar coordinates?

To graph polar coordinates, plot the distance from the origin (radius) on the y-axis and the angle from the reference line (polar axis) on the x-axis. Then, use the angle to determine the direction and the radius to determine the distance from the origin.

3. What is an integral in polar coordinates?

In polar coordinates, an integral is a mathematical operation that calculates the area under a curve in a polar graph. It is represented by the symbol ∫ and is used to find the area of irregularly shaped regions in polar coordinates.

4. Can you provide an example of solving an integral in polar coordinates?

Yes, for example, an integral in polar coordinates could be ∫0 4r2 dθ, which represents the area under the curve of a circle with a radius of 2. To solve this, we would use the formula for the area of a circle (A = πr2) and integrate it with respect to θ, resulting in an answer of 16π.

5. How are polar coordinates used in real life?

Polar coordinates are commonly used in fields such as physics, engineering, and navigation. They can be used to describe the position and movement of an object in a circular or rotational motion, such as the orbit of planets or the rotation of a wind turbine. They are also used in mapping and location-based applications, such as GPS, to determine the distance and direction from a fixed point.

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