Evaluate a definite integral using the Fundamental Theorem of Calculus
Informally
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Differentiation and (definite) integration are inverse operations.Slope is a quotient. The limit of the slope of secant lines gives the slope of a tangent line.Area is a product. The limit of the area of rectangles gives the area of a region under a curve.Quotients and products are inverse operations (division and multiplication).
Theorem 4.9 The Fundamental Theorem of Calculus
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If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], thenS_a^b f(x) dx = F(b) - F(a).
Guidelines for using the Fundamental Theorem of Calculus
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See page 283 in your text.Note especially item 3: You do not need a constant of integration when working with definite integrals.
Understand and use the Mean Value Theorem for Integrals
Theorem 4.10 Mean Value Theorem for Integrals
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If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that S_a^b f(x) dx = f(c)(b-a).
Find the average value of a function over a closed interval
Definition of the average value of a function on an interval
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If f is integrable on the closed interval [a,b], then the average value of f on the interval is 1/(b-a) S_a^b f(x) dx.
Understand and use the Net Change Theorem
Theorem 4.12 The Net Change Theorem
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The definite integral of the rate of change of a quantity F'(x) gives the total change, or net change, in that quantity on the interval [a,b].S_a^b F'(x) dx = F(b) - F(a) Net change of F
Understand and use the Second Fundamental Theorem of Calculus
The Definite Integral as a Function of x
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The definite integral as a numberS_a^b f(x) dxThe definite integral as a function of xF(x) = S_a^b f(t) dt
Accumulation Function
Theorem 4.11 The Second Fundamental Theorem of Calculus
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If f is continuous on an open interval I containing a, then, for every x in the interval,d/dx [ S_a^x f(t) dt ] = f(x).
