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The purpose of determining an integral is to find the exact area under a curve or the exact value of a function in a given interval. This can be useful in many applications, such as calculating the displacement of an object or finding the total amount of a substance in a chemical reaction.
An integral is calculated by finding the antiderivative of a function and then evaluating it at the upper and lower limits of the integral. This process is known as the Fundamental Theorem of Calculus.
A definite integral has specific upper and lower limits, while an indefinite integral does not. The result of a definite integral is a single numerical value, whereas the result of an indefinite integral is a function with a constant added.
Yes, an integral can have a negative value if the function being integrated has negative values in the given interval. The integral simply represents the net signed area under the curve, so if there is more area below the x-axis than above it, the integral will be negative.
Integrals are commonly used in physics, engineering, economics, and other fields to solve problems involving rates of change, area, volume, and optimization. They can also be used to find the center of mass of an object, calculate work done by a force, and determine the average value of a function.
