What shape results from integrating the area of a circle?

[Rob K]

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:
$\int2∏r$ = ∏r$^{2}$

but what do I get if I integrate the area, I get
∏r$^{3}$/3

I am confused as to what this shape would be, I kind of was expecting a sphere, but the formula for a sphere is:
4∏r³/3

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

 

[LCKurtz]

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:
$\int2∏r$ = ∏r$^{2}$

but what do I get if I integrate the area, I get
∏r$^{3}$/3

I am confused as to what this shape would be, I kind of was expecting a sphere, but the formula for a sphere is:
4∏r³/3

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

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.

 

[Rob K]

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?

 

[LCKurtz]

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?

Yes, that also describes a right isosceles cone. Both legs of the triangle have length r.

 

[mathman]

The area of the surface of a sphere is 4πr². Integrate that to get the volume of a sphere.

 

[Rob K]

Thank you.