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

Area under Parabolas | Differential Calculus | SNS Institutions

#snsinstitutions #snsdesignthinkers #designthinking The area under a parabola is calculated using the method of definite integration because a parabola represents a curved boundary that cannot be measured accurately using simple geometric formulas alone. A parabola is generally represented by a quadratic function. The graph of this function forms a symmetric curve (in standard cases) that opens either upward or downward depending on the value. To determine the area under a parabola between two points on the x-axis, the curve is divided into infinitely small vertical strips. Each strip has a small width and a height equal to the value of the function at that point. By summing the areas of these infinitesimally small strips through integration, we obtain the exact area bounded by the curve and the x-axis within the given limits. If the parabola lies above the x-axis within the interval, the definite integral directly gives the positive area. If part of the parabola lies below the x-axis, the integral gives a negative value for that portion, and the absolute value must be taken to calculate the total physical area. The limits of integration are usually the points where the parabola intersects the x-axis or any other specified boundary. A special theoretical result for a standard parabola states that the area enclosed between a parabola and a straight line (such as its base) is equal to two-thirds of the area of the rectangle formed by the base and the maximum height of the parabola. This property highlights the unique geometric nature of parabolic curves. The concept of finding the area under parabolas is important in engineering and physics. It is used in problems involving projectile motion, structural arches, satellite dishes, and various mechanical components where parabolic shapes naturally occur. Integration therefore provides a precise and systematic way to measure areas bounded by parabolic curves.