A definite integral is a mathematical concept that represents the area under a curve on a graph. It is denoted by the symbol ∫ and has a lower and upper limit that define the range of values to be integrated.
To calculate a definite integral, you first need to find the anti-derivative of the function. Then, you can substitute the upper and lower limits into the anti-derivative and subtract the two values to find the area under the curve.
The relationship between a definite integral and a limit is that the definite integral can be seen as the limit of a Riemann sum, which is a way of approximating the area under a curve by dividing it into smaller rectangles. As the number of rectangles increases to infinity, the Riemann sum approaches the value of the definite integral.
To prove a relation using a definite integral and a limit, you need to show that the limit of the Riemann sum is equal to the definite integral. This can be done by choosing a specific function and limits, and then evaluating both the definite integral and the limit to show that they are equal.
Definite integrals and limits have many real-life applications, such as calculating the distance traveled by an object with a changing velocity, finding the volume of irregularly shaped objects, and determining the average value of a function over a certain interval. They are also used in economics, physics, and engineering to model and analyze various systems.