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The Area Under a Curve

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chrisheaven

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The Area Under a Curve
Can someone say something about this mathemaically?
 
Go read the chapters of a calculus book about integrals. 😀
 
That was known to be 1/3 by Archimedes of Syracuse long, long before Newton was born. 😀
 
That I did not know. 😀

He knew that the area under y=x^n between x=0 and x=a was (a^(n+1))/(n+1). Quite a remarkable result before calculus, but he was a genius.
 
No, the only time I was ever in the UK I was only in London. I've never been to Bath. 😀
 
The definite integral can be used to find the area between a graph curve and the ‘x’ axis, between two given ‘x’ values. This area is called the ‘area under the curve’ regardless of whether it is above or below the ‘x’ axis.

When the curve is above the ‘x’ axis, the area is the same as the definite integral ...
 
Well, you've gotten this thread back on topic. 😀
In applications in economics, a negative area can have a real meaning, a loss rather than a profit.
 
A double integral can give the volume under a surface. 😀
 
The double integral sign is used to indicate mathematical integration performed twice in succession.
Double integrals are used to calculate the integrals of functions in two variables.
An example is the volume under a surface in three-dimensional (3D) space.
The double integral sign appears as two single integral signs, placed one after the other
 
It was Guido Fubini who proved that, under many common circumstances, a double integral will give the same answer in either order of integration. 😀
 
Guido Fubini was born in Venice in 1879. He was urged toward mathematics by his father, a teacher in a mechanics school. At age 19, he enrolled in the Scuola Normale Superiore in Pisa, where he obtained his mathematics doctorate in 1900 at age 21. His thesis, published in 1902, was in geometry.

Fubini's teaching career developed rapidly. He taught first at the University of Catania in Sicily, a bit later at the University of Genoa, and in 1908 he went to the University of Turin. During the First World War, he became engaged with military problems that led him to do applied mathematics and mathematical physics.

When the events that led to World War II were unfolding, Fubini had to consider what was best for his wife and two engineer sons. When the Institute for Advanced Study in Princeton offered him a position in 1939, he accepted and emigrated to the United States.

His worked in many areas of mathematics, but his most important research centered on what was called absolute differential calculus, now known as tensor analysis, which is a method of studying quantities that depend linearly on vectors. And of course you, and many others, know of him because of Fubini's theorem.

Already in poor health when he left Italy, he died after about four years in America.
 
chrisheaven said:
When the events that led to World War II were unfolding, Fubini had to consider what was best for his wife and two engineer sons. When the Institute for Advanced Study in Princeton offered him a position in 1939, he accepted and emigrated to the United States.
Click to expand...

This was a non-trivial consideration since his wife was Jewish (and hence also his sons half-Jewish under the racial laws of his country's ally, Germany). His family's decision to come to the United States may have saved those three lives.
 
Yes, a very wonderful outcome for his wife and two sons. 😀
 
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