MAT.126 4.4

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MAT.126 4.4 The Fundamental Theorem of Calculus

Understand and use the Second Fundamental Theorem of Calculus

Theorem 4.11 The Second Fundamental Theorem of Calculus

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).

Accumulation Function

The Definite Integral as a Function of x

The definite integral as a number

S_a^b f(x) dx

The definite integral as a function of x

F(x) = S_a^b f(t) dt

Understand and use the Net Change Theorem

Theorem 4.12 The Net Change Theorem

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

Find the average value of a function over a closed interval

Definition of the average value of a function on an interval

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 Mean Value Theorem for Integrals

Theorem 4.10 Mean Value Theorem for Integrals

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).

Evaluate a definite integral using the Fundamental Theorem of Calculus

Guidelines for using the Fundamental Theorem of Calculus

See page 283 in your text.

Note especially item 3: You do not need a constant of integration when working with definite integrals.

Theorem 4.9 The Fundamental Theorem of Calculus

If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

S_a^b f(x) dx = F(b) - F(a).

Informally

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).