- #1
erocored
- 30
- 7
I know he had this ratio:
But how did he get this:
?
But how did he get this:
B How did Cavalieri get his formula for the area underneath a parabola?
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Cavalieri derived his formula for the area underneath a parabola by utilizing a specific ratio related to the sum of squares. The key formula involved is the sum of the first m squares, expressed as $$\sum_{k=1}^m k^2 = \frac{m(m + 1)(2m + 1)}6$$. By substituting this sum into his initial equation, he was able to arrive at the desired result. This mathematical approach highlights the connection between geometric areas and algebraic summation. Understanding this derivation is crucial for grasping Cavalieri's contributions to calculus and geometry.
- #2
erocored
- 30
- 7
- #3
Mark44
Mentor
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- 10,572
The result uses a formula for the sum of squares.erocored said:
$$\sum_{k=1}^m k^2 = \frac{m(m + 1)(2m + 1)}6$$
Replace ##1^2 + 2^2 + \dots + m^2## in your first equation by the above, and you will get the result shown in the second equation.
Hello!
I tried to calculate projections of curl using rotation of coordinate system but I encountered with following problem.
Given:
##rot_xA=\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}=0##
##rot_yA=\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}=1##
##rot_zA=\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}=0##
I rotated ##yz##-plane of this coordinate system by an angle ##45## degrees about ##x##-axis and used rotation matrix to...
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