Area under curves | Differential Calculus | SNS Institutions скачать в хорошем качестве

Area under curves | Differential Calculus | SNS Institutions

#snsinstitutions #snsdesignthinkers #designthinking The concept of finding the area under a curve using integration is one of the fundamental applications of integral calculus. When a function is represented graphically, the region between the curve and the x-axis forms a measurable area. Integration provides a systematic and accurate method to calculate this area, especially when the boundary is curved and cannot be determined using simple geometric formulas. The basic idea behind integration comes from dividing the area under a curve into many small rectangles. Each rectangle has a small width along the x-axis and a height determined by the value of the function at that point. By adding the areas of all these rectangles, we obtain an approximate value of the total area. When the width of these rectangles becomes extremely small, the approximation becomes exact. This limiting process is called definite integration. If a function is continuous over a given interval, the definite integral of the function between two limits gives the exact area under the curve between those limits. The definite integral not only calculates area but also considers the position of the curve relative to the x-axis. If the curve lies above the x-axis, the area is considered positive, and if it lies below, the area is considered negative. To find the total physical area, the absolute value is taken where necessary. The concept is based on the Fundamental Theorem of Calculus, which connects differentiation and integration. It states that if a function has an antiderivative, the definite integral between two limits can be found by evaluating the antiderivative at those limits and taking their difference. This theorem makes the calculation of area more practical and efficient. Thus, integration provides a powerful mathematical tool for determining the area of irregular shapes bounded by curves. It is widely used in engineering, physics, and economics to measure quantities such as displacement, work done, and accumulated change.