The definite integral represents the area under a curve on a specific interval from "summary" of Skills in Mathematics - Integral Calculus for JEE Main and Advanced by Amit M Agarwal
The concept of the definite integral is based on the idea that it represents the area under a curve on a specific interval. When we talk about the area under a curve, we are essentially looking at the region enclosed by the curve and the x-axis within a given interval. This region can be positive, negative, or zero, depending on the nature of the curve and the interval in question. To understand this concept better, let's consider a simple example. Suppose we have a curve defined by a function f(x) on the interval [a, b]. The definite integral of f(x) from a to b, denoted by ∫[a, b] f(x) dx, gives us the net area between the curve and the x-axis over the interval [a, b]. This net area takes into account the portions above and below the x-axis, with appropriate signs. It is important to note that the definite integral can be used to calculate the area under a curve even when the curve dips below the x-axis. In such cases, the area below the x-axis contributes negatively to the overall net area. This is why the definite integral is such a powerful tool in calculus, as it allows us to handle both positive and negative areas with ease. The concept of the definite integral representing the area under a curve on a specific interval is fundamental in calculus and is used in various applications, such as calculating the total distance traveled by an object, finding the total accumulated change in a quantity, or determining the total work done by a force over a given interval. By understanding this concept and mastering the techniques of integration, we can solve a wide range of problems in mathematics and science that involve quantities that vary continuously.
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