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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Also their sum does add up to a definite value 0 b fx

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Unformatted text preview: tain volume of water, or charge a capacitor from the given START TIME. 3. If we know the STOP TIME and CHANGE in VOLUME, then we may determine the START TIME for the given RATE of FLOW f ’ (t) . 166 32. INDEFINITE and DEFINITE INTEGRAL Let f(x) = x2. When we differentiate f(x) to get the derivative f’(x) = 2x, we do differentia deriv tiv dif entiate deri not use the term “indefinite” to say : f’(x) is the “indefinite derivative” of f(x). “indefinite” “indefinite deriv tiv “indef “indef Also, when we evaluate f’(x) = 2x at some instant a to get 2a , we do not use the 2a “definite” “definite deriv tiv term “definite” to say : 2a is the “definite derivative” of f(x) at x = a. “def 2a “def When it comes to integration we have the terms indefinite and definite. integ indefinite definite inte indef def F(x) = ANTIDERIVATIVE {f(x)} = ∫ f (x)dx is called the INDEFINITE INTEGRAL of the integrand f(x). The INDEFINITE INTEGRAL F(x) of a given function f(x) is integrand the most general form of its ANTIDERIVATIVE. We may evaluate the INDEFINITE INTEGRAL F(x) over and interval [a, b]. b ∫ a f(x)dx = [F(x)]ab = F(b) − F(a) which is the CHANGE in F(x) from a to b. b ∫ a f (x)dx is also called the DEFINITE INTEGRAL of the integrand f(x). integrand We may evaluate FC (x) = F(x) + C over and interval [a, b]. In the process the CONSTANT OF INTEGRATION C disappears. b ∫ a f(x)dx = [FC (x) ]ab = { F(b)+ C } − { F(a) + C } = F(b) − F(a) which is again only the CHANGE in FC (x) from a to b. If F(x) is the expression of CHANGE in position , then evaluation of the INDEFINITE position INTEGRAL F(x) over the inter v al [ a, b] is only the CHANGE in position i nter p osition from a to b = F(b) − F (a). To find the position at any chosen instant xi starting from instant a, we need position instant to know F(a), the the position at instant a, and the CHANGE or DEFINITE INTEGRAL xi ∫a f (x)dx. 167 y(t1) y(t2) h1 321 321 321 321 321 321 321 (0,0) t1 vertical posit...
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