The area under a curve refers to the numerical value of the region bounded by a curve and the x-axis on a graph. It is a measure of the total space occupied by the curve within a specific interval.
Finding the area under a curve is important in various fields of science and mathematics. It helps in solving real-world problems involving rates of change, such as velocity and acceleration. It is also used in calculating probabilities and in determining the total value of a quantity over a given interval.
The process of finding the area under a curve involves dividing the curve into small, equal intervals and approximating the area under each interval using geometric shapes such as rectangles or trapezoids. Then, the sum of these approximated areas is calculated to get a more accurate estimation of the total area under the curve.
The most common methods used to find the area under a curve are the Riemann sum, the Trapezoidal rule, and Simpson's rule. These methods involve dividing the curve into smaller intervals and using different formulas to approximate the area under each interval.
Finding the area under a curve has various real-life applications, such as in calculating the distance traveled by an object given its velocity function, determining the amount of medicine in a patient's bloodstream, and estimating the total energy consumption of a household over a period of time.