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Source: http://mathworld.wolfram.com/PappussCentroidTheorem.html
Since the generated axis, defined as x is equal to 1/2(r^2) , then we can say
2/3(r^3) = 1/2A(r^2)
and therefore,
(2/3)(2)[(r^3)/(r^2)] = A
(4/3)(r/ ) = A
The Calculus
The equation V = 2Ax can be interpreted as the accumulated volume of a function B(x) on the interval [a,b] to form a solid. This is because the volume of a solid is
(B(x))^2dx [a,b] = V = 2Ax .
This is true because the integral multiplied by pi is the accumulator of the volume of a function, and (B(x))^2 is the area of the section of the solid that is being accumulated. In the case of Pappus' centroid theorem, 2x is the path that the Area of the section is traveling.
Essentially, this means the Area of the lamina multiplied by its generated lamina is equal to 2/3(r^3).
When we are finding the volume of a solid of revolution we are accumulating the area of the solid using an axis to rotate about. Therefore, we must determine what shape is being used to accumulate the area of the volume. Since we are finding the area of a sphere, it would be logical to use a semicircle, because we may then use the segment to rotate it about its diameter to find the volume. A semicircle is equal to 1/2(r^2) .
The volume V of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area (A) of the lamina and the distance (d) traveled by the lamina's geometric centroid (x).
V = A*d = 2Ax
Lamina - A section of the 3D shape that can be used to accumulate area.
Geometric Centroid - The center of mass of a two-dimensional planar lamina or a three-dimensional solid
The volume of a sphere is 4/3(r^3)0 .
This volume can be found using the Pappus' centroid theorem if you set V = 4/3(r^3) = 2 Ax
Since 4/3(r^3) = 2Ax
We can say
4/3(r^3) = 2 Ax
and therefore
2/3(r^3) = Ax