The definite integral- area under a curve, Mathematics

The Definite Integral

Area under a Curve

If there exists an irregularly shaped curve, y = f(x) then there is no formula to find out the area under the curve between two points x = a and x = b on the horizontal axis. If this interval [a, b] is broken into 'n' subintervals [x1, x2], [x2, x3] ... [xn-1, xn] and rectangles are constructed in such a way that the height of each rectangle is equal to the smallest value of the function in the subinterval then the sum of the areas of the rectangles i.e.  158_area under the curve.png will approximate the actual area under the curve, where  642_area under the curve1.png , is the difference between any two consecutive values of x. The smaller the value of  642_area under the curve1.png the more rectangles can be created and the closer is the sum of the areas of the rectangles so formed, i.e.  158_area under the curve.png , to the actual area under the curve. If the number of subintervals increases, that is 'n' approaches infinity, each subinterval becomes infinitesmally small and the area under the curve can be expressed as

Area, C = 778_area under the curve2.png

Figure 1

435_area under the curve3.png

Figure 2

379_area under the curve4.png

The area under the graph of a continuous function between two points on the horizontal axis, x = a and

x = b, can be best described by the definite integral of f(x) over the interval x = a to x = b. This is mathematically expressed as

1832_area under the curve5.png 

a and b on the left hand side of the above expression are called the upper and lower limits of the integration. Unlike the indefinite integral which represents a family of functions as it includes an arbitrary constant, the definite integral is a real number which can be found out by using the  = 

fundamental theorem and is expressed as  1298_area under the curve6.png
Posted Date: 9/13/2012 7:50:13 AM | Location : United States







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