Area Under a Curve Formulating the area under a curve is the first step toward developing the concept of the integral. The area under the curve formed by plotting function f(x) as a function of x can be approximated by drawing rectangles of finite width and height f equal to the value of the function at the center of the interval. Show area integral for simple geometries.Show area approximation of an integral. 
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Integral as Limit of Area The approximation to the area under a curve can be made better by making the approximating rectangles narrower. The idea of the integral is to increase the number of rectangles N toward infinity by taking the limit as the rectangle width approaches zero.
While the concept of geometrical area is a convenient way to visualize an integral, the idea of integration is much more general than that. Any continuous physical variable can be "chopped up" into infinitesmal increments (differential elements) so that the sum of the product of that "width" and some function approaches an infinite sum. The integral is a powerful tool for modeling physical problems which involve continuously varying quantities.
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Area Integral ExamplesArea examples with simple geometry can reinforce the idea of the integral as the area under a curve. For a function which is just a constant a, then the area formed by the function is just a rectangle.
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Area Integral Approximations The area under any continuous curve can be approximated by drawing a number of rectangles. The integral is the limit for an infinite number of rectangles. Show the integral which is the limiting case. 
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Area Integral ExampleIntegrals are useful for finding the area under curves which can be approximated by geometrical methods.
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