Triangle Inequality, Integrals

In summary, the Triangle Inequality Theorem is a principle in geometry that states the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. It is commonly used to determine if a set of line segments can form a triangle. An integral is a mathematical concept used to find the area under a curve on a graph and the total value of a function over a specific interval. The fundamental theorem of calculus states that differentiation and integration are inverse processes, with the first part relating derivatives to integrals and the second part relating definite integrals to antiderivatives. Integrals have various real-life applications in fields such as physics, engineering, economics, and statistics, and are used to calculate quantities such
  • #1
psholtz
136
0
Is it true in general that:

[tex]|\int f(x)dx| < \int |f(x)|dx [/tex]

Not sure if "Triangle Inequality" is the right word for that, but that seems to be what's involved.
 
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  • #2
It is true in general simply because -|f(x)|≤f(x)≤|f(x)|.
 

What is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

How is the Triangle Inequality Theorem used in geometry?

The Triangle Inequality Theorem is used to determine whether a set of three given line segments can form a triangle or not. If the sum of the lengths of two sides is greater than the length of the third side, then a triangle can be formed.

What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to calculate the total value of a function over a specific interval.

What is the fundamental theorem of calculus?

The fundamental theorem of calculus states that differentiation and integration are inverse processes of each other. It has two parts, the first part relates derivatives to integrals, and the second part relates definite integrals to antiderivatives.

How are integrals used in real-life applications?

Integrals are used in various fields such as physics, engineering, economics, and statistics to calculate important quantities such as displacement, velocity, work, and probability. They are also used to solve optimization problems and to find areas and volumes of irregular shapes.

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