Integration is a mathematical technique used to find the area under a curve or the accumulation of a quantity over time. It is important in calculus because it allows us to solve problems involving rates of change, such as finding the distance traveled by an object or the total amount of a substance produced over time.
The two main methods of integration are the definite integral, which involves finding the exact value of the area under a curve, and the indefinite integral, which involves finding a function whose derivative is equal to the original function. Other methods include integration by parts, substitution, and trigonometric substitution.
Limits are used in calculus to describe the behavior of a function as the input approaches a certain value. They allow us to determine the value of a function at a point where it is not defined, or to find a function's maximum or minimum values. Limits are also used to define derivatives and integrals.
The average value of a function over an interval is found by taking the definite integral of the function over that interval and dividing it by the length of the interval. This gives the average height of the function over the interval, which can be thought of as the average value.
Yes, integration is used in many real-life applications, such as calculating the area under a demand curve in economics, finding the volume of irregularly shaped objects in physics, and determining the amount of medication in a patient's bloodstream over time in medicine. It is a powerful tool for solving real-world problems involving rates of change.