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Unit4 - Unit 4 The Denite Integral We know what an indenite...

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Unit 4 The Definite Integral We know what an indefinite integral is: the general antiderivative of the integrand function. There is a related (although in some ways vastly differ- ent) concept, the definite integral, which uses similar-looking notation. The definite integral of a function is defined only for intervals on which the func- tion is continuous. Let’s briefly review what that means. Recall the definition of continuity of a function at a point: Definition 4.1. A function f ( x ) is said to be continuous at a value c in the domain of f if lim x c f ( x ) = f ( c ) Note that this requires that: 1. f ( c ) must be defined, 2. lim x c f ( x ) must exist, (i.e., the function approaches the same limiting value from both sides at c .) 3. and these two numbers must be equal. Also recall how we extend this idea of continuity to intervals: Definition 4.2. A function f ( x ) is said to be continuous on a closed interval [ a, b ] if f is continuous at c for every value c [ a, b ]. In layman’s terms, a function is continous on an interval if you can draw the function without having to lift your pencil off the page. 1
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Discontinuities in a function occur at two kinds of places: 1. places where the function is not defined (for instance a place where the function has denominator = 0,) 2. places where the function suddenly jumps from one value to another. Consider any function f ( x ) which is continuous on some interval [ a, b ]. The function may be positive-valued in some parts of the interval and negative- valued in other parts of the interval. That is, the graph of y = f ( x ) may lie above the x -axis in some places and below the x -axis in others, all within the interval [ a, b ]. Definition 4.3. (Preliminaries for definition of definite integral) Consider a function f which is continuous on some interval [ a, b ]. Let R + be the region or regions which lie below the curve y = f ( x ) and above the x -axis (i.e., regions in which the function is positive-valued) within the interval [ a, b ]. Similarly, let R - be the region or regions which lie above the curve y = f ( x ) and below the x -axis (i.e., regions in which the function is negative-valued) within the interval [ a, b ]. Finally, let A ( R + ) and A ( R - ) denote the (total) areas of these regions, re- spectively. The definite integral of the function f ( x ) from x = a to x = b is defined as the difference between these two areas, i.e., the net area above the x -axis , on the interval [ a, b ]. We have: Definition 4.4. Let f be any function and let [ a, b ] be any finite closed interval such that f is continuous on [ a, b ]. Let A ( R + ) and A ( R - ) be defined as above. The symbol integraltext b a f ( x ) dx , called the definite integral of f ( x ) from a to b , is defined as: integraldisplay b a f ( x ) dx = A ( R + ) - A ( R - ) Notice: Unlike an indefinite integral, which is a function, a definite integral is a number . 2
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For some (only a few) functions, the regions R + and R - have shapes which allow their areas to be calculated easily (e.g. regions whose shapes are rectangles, triangles, trapezoids, semi-circles, etc.). However, for the vast majority of functions, finding the areas of these regions is not so straightfor- ward. (In a more rigorous study of calculus, such an area would be found
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