Polar Coordinates and Area of a Circle

Related BrainMass Solutions

• Polar Coordinates : Circles and Integrals

  a) in polar coordinates, write equations for the line x=1 and the circle of radius 2 centered at the origin b) write an integral in polar coordinates representing the area of the region to the right of x=1 and inside the circle c) evaluate the integral

• the trigonometric identity

  The polar equation is: (1.1) Then: (1.2) Using the polar coordinates transformation: (1.3) Equation (1.2) becomes: (1.4) This is an equation of a circle of radius 1 centered at Since in our case the angle is defined as , this

• Volume of a solid

  We can change the variable x to so now the circle is centered at (0,0) and the volume element is now: And now, the area of integration is a circle centered at (0,0) with radius 2.

• Double Integral Evaluated

  Then So , and it the whole circle, In the polar coordinates, So This provides an example of evaluating a double integral using polar coordinates.

• Revolutions of integrals - torus

  When C is a circle, the surface obtained is a circular torus or torus of revolution The polar coordinates of circle indicate that the radius of circle is a.

• position of the center of mass of a thin parabolical shell

  We know that: z'*pi*a^2 = (a-z')*2*pi*a^2;(this comes from: area of circle=4pi*R^2) z' = -2z'+2a; z'= 2*a/3 This specifies the position of the center of mass The center of mass of a thin parabolic shell is figured.

• Plot the point whose polar coordinates are given.

  Then find two other pairs of polar coordinates of this point, one with r>0 and one with r<0 Points are (-1, pi/5) The top two graphs show the differences between Cartesian and polar coordinates.

• Evaluating the Given Integral by Changing to Polar Coordinates

  For the upper part, for each , we also get a circle which satisfies . In the polar system, we can suppose , and , where and , .

• Green's Theorem and Line Integrals

  Therefore, it will be easier to use polar coordinates: Therefore we get: The second curve where defines an area of a triangle with vertices at (1,0), (0,1) and (0,0).