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Integration as a Process of Summation (using integration to calculate the area under a curve between two x values)

Consider the area under a curve • The area under a curve can be thought of as a series of strips with width and height y (=f(x)) • So the area of one strip is: x xy.

Consider the area under a curve • Now consider the graph: • The total area between the boundary values of x=a and x=b is: • Since the strip is an approximation of the area under the curve, the area of the strip approaches the area under the curve as • So, b a xy. A 0 x b ax xy. A 0 lim

Consider the area under a curve • Now consider an alternative expression for A: • Considering the same starting point of: • Then: • Again this approximation becomes more accurate as: • So, xy. A 0 x y x. A x 0 lim y x.

Consider the area under a curve • So, • But, • So between x=a and x=b: y x. A x 0 lim ydx. ATherefore y dxd. A So dxd. A x , , lim 0 b a ydx.

Consider the area under a curve • Putting the two alternative equations together: • And therefore integration is a process of summation. b ab ax ydxxy. A 0 lim