Definite Integral: Limit of a Summation

In summary, a definite integral is a mathematical concept that represents the area under a curve between two points on a graph. It is calculated by taking the limit of a summation, where the width of the rectangles used to approximate the area becomes smaller and smaller. This concept is fundamental in calculus and has various applications in real-life situations, such as calculating volumes and areas in physics and engineering. To calculate it, one must find the indefinite integral or antiderivative of the function and plug in the upper and lower limits of integration. The definite integral is commonly used in various fields, including physics, engineering, economics, and statistics, to solve problems involving rates of change and accumulation.
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Homework Statement



Hi guys, i have a exercise of the limit of a summation that is the formal definition of definite integral and i need resolve and explain, but i can't resolve for the rational exponent, for this, need help, thanks in advance.

Homework Equations



[itex]\lim_{n \rightarrow \infty} \sum_{i=1}^{n} {(1+\frac{2}{n}(i-0.3))^{\frac{7}{5}}\frac{2}{n}[/itex]

The Attempt at a Solution



I can solve this expretion but with a integer exponent, not with a rational exponent.
 
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  • #2
Help me, please.
 
  • #3
That really doesn't look like a Riemann sum to me. Were you given that sum?
 

FAQ: Definite Integral: Limit of a Summation

What is the definition of a definite integral?

A definite integral is a mathematical concept that represents the area under a curve between two points on a graph. It can be thought of as the limit of a summation, where the width of the rectangles used to approximate the area becomes smaller and smaller, resulting in a more accurate calculation of the area.

How is the definite integral related to the limit of a summation?

The definite integral is the limit of a summation because it is calculated by taking the limit as the width of the rectangles used in the summation approaches zero. This allows for a more precise calculation of the area under the curve.

What is the importance of the definite integral in calculus?

The definite integral is a fundamental concept in calculus and is used to find the area under a curve, as well as to solve problems involving rates of change and accumulation. It is also used in many real-life applications, such as calculating volumes and areas in physics and engineering.

How do you calculate the definite integral of a function?

To calculate the definite integral of a function, you first need to find the indefinite integral or antiderivative of the function. Then, you can plug in the upper and lower limits of integration into the antiderivative and subtract the result at the lower limit from the result at the upper limit.

What are some common applications of the definite integral?

The definite integral is used in many areas of mathematics and science, including physics, engineering, economics, and statistics. It is used to calculate areas, volumes, and averages, as well as to solve problems involving rates of change and accumulation.

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