Question

Use polar coordinates to find the formula for the area A of the ball

B²ᵣ = {(x,y) ∈ R² ∣ x² +y² ≤ r²}

Answer 1

The area A of a ball in polar coordinates can be represented by the formula A = πr², the same as in Cartesian coordinates because the area is independent of the coordinate system.

Step-by-step explanation:

The student is asking to use polar coordinates to find the formula for the area of a ball, defined as B²ᵣ, which represents the set of points (x, y) in R² where the sum of the squares of x and y is less than or equal to r², essentially a circle with radius r.

In polar coordinates, the relationship between the Cartesian and polar systems can be described by the equations x = r × cos(φ) and y = r × sin(φ), where r is the radial distance from the origin to a point, and φ is the angle between the line connecting this point to the origin and the positive x-axis.

The area A of a circle in Cartesian coordinates is given by A = πr². By applying the concept of polar coordinates, we can establish that the same formula represents the area in polar coordinates since the circle's area does not depend on the coordinate system used.

Here's a step-by-step explanation to derive the formula for the area using polar coordinates:

1. 
   
3. 
   
5. 
   
7.