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Integration of circle

  1. Jan 6, 2012 #1
    Hi there,

    I am trying to understand calculus as concerns circles and I can clearly see that the integral of a circumference is an area:
    [itex]\int2∏r[/itex] = ∏r[itex]^{2}[/itex]

    but what do I get if I integrate the area, I get

    I am confused as to what this shape would be, I kind of was expecting a sphere, but the formula for a sphere is:

    plus a little technical point: when differentiating this, is it dr/dx or dy/dr, or am I totally off the mark?

    Thanks in advance

    Rob K
  2. jcsd
  3. Jan 6, 2012 #2


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    The integration is with respect to r. The volume you get is a right isosceles cone with base and height both r. Here's how you can think of it. Take a circle of radius r and think of an r axis perpendicular to the circle through its center. The circle is at distance r from the origin on this axis. If you slide the circle to a larger distance its radius increases accordingly. Sliding the circle in the r direction generates the cone and the integral you are calculating represents calculating the volume of that cone by circular cross sections. And it gets the correct answer of 1/3*Area of base*height for a cone.

    The connection between circumference and area of a circle by integration also works for a sphere, but the connection is between surface area and volume. The surface area is ##4\pi r^2## and if you integrate that you get the volume ##\frac 4 3 \pi r^3##. This is the calculation of the volume of a sphere by spherical shells.
  4. Jan 6, 2012 #3
    Thank you, that is very useful as a visualization of integration. Let me get this a little clearer in my head. Is this another way to describe it. a right angle triangle with two 45˚ angles then revolved around the z axis, assuming the z axis is one of the non hypotenuse sides, as you would create it in 3d modelling?
  5. Jan 6, 2012 #4


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    Yes, that also describes a right isosceles cone. Both legs of the triangle have length r.
  6. Jan 6, 2012 #5


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    The area of the surface of a sphere is 4πr2. Integrate that to get the volume of a sphere.
  7. Jan 6, 2012 #6
    Thank you.
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