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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Direction integ direction of integr ation and change

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Unformatted text preview: that f(x) = F ’ (x). F(x) = ANTIDERIVATIVE of {f(x)}. And once we prove this, we call f(x) the integrand and F(x) the integral . integrand integral integrand = the operand of the integration operation. integral = ∫integrand . Thus our Table of DERIVATIVES and ANTIDERIVATIVES becomes a Table of Integrals. Integ als. 135 27. F(x ANTIDERIVATIVE{f(x)} 27. F( x ) = ANTIDERIVATIVE {f(x)} = ∫ f(x)dx f(x)dx Let us look at two examples to see the relation between : F(x) = ar ea under the cur ve of f(x) = ∫ f(x)dx area curv f(x)dx F(x ANTIDERIVA {f(x) f(x)} F( x ) = ANTIDERIVATIVE {f(x)} . and Example 1: Let f(x) = 1. For b = 1, 2, 3, . . . we may compute the area function b F(x) = ∫ f(x)dx and plot the graph of F(x). f(x)dx 0 2 0 f(x) = 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432187654321 1 9 98765432987654321 1 1 2 b F(x) = ∫ f(x)dx f (x)dx 0 1 2 1 0 x =x 1 2 x d F(x) From the diagrams above it is clear that F(x) = x and f(x) = = 1. dx Example 2: Let f(x) = x. For x = 1, 2, 3, . . . we may compute the area function b F(x) = ∫ f(x)dx and plot the graph of F(x). f(x)dx 0 4 f(x f(x) = x 0987654321 1 098765432987654321 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 2 1 2 F(x) = ∫ f(x)dx = x f(x)dx 2 0 2 3 22 2 2 1 0 b 1 x 0 12 2 1 2 3 4 x d F(x) From the diagrams above it is clear that F(x) = 1/2 x 2 and f(x) = = x. dx 136 From these two particular examples it is reasonable to infer that : area ANTIDERIVA ar ea function F(x) = ANTIDERIVATIVE {f(x)}. Proof Pr oof : Let f(x) be the cur ve between x = a and x = b with f(a) = A and f(b) = B. Divide the interval [a,b] into n sub-intervals ∆ x i and construct rectangles in step-like fashion as shown in the figure. Let F(x) = the area under the cur ve of f(x). area curv ar B f(x) 4321 4321 4...
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