FTC Review; The Method of Substitution

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Presentation transcript:

FTC Review; The Method of Substitution February 4, 2004

The Definite Integral as Area Let f be a continuous function defined on the interval [a, b]. The definite integral of f from a to b, denoted by represents the total signed area of the region bounded by y = f (x), the vertical lines x = a and x = b, and the x-axis.

Properties of Definite Integrals Let f and g be continuous functions defined on the interval [a, b]. Furthermore, let c and k be constants such that a < c < b. Then…

The Fundamental Theorem of Calculus Let f be a continuous function defined on [a, b], and let F be any antiderivative of f. Then

Keeping It Straight Definite Integral Area Function Represents a real number (a signed area). Area Function Represents a single antiderivative of f. Indefinite Integral Represents the entire family of antiderivatives of f.

Substitution Rule for Indefinite Integrals

Implementing the Substitution Rule Choose u. Differentiate u w.r.t. x and solve for du. Substitute u and du into the old integral involving x to form a new integral involving only u. Antidifferentiate with respect to u. Re-substitute to find the antiderivative as a function of x.

Two Special Forms

Substitution Rule for Definite Integrals

Implementing the Substitution Rule (Definite Integrals) Choose u = g(x). Differentiate u w.r.t. x and solve for du. Substitute u and du into the old integral involving x, as well as converting endpoints from a and b to g(a) and g(b). Antidifferentiate with respect to u and evaluate at the new endpoints.

Arcsine (Inverse Sine Function) For x in [-1, 1], y = arcsin x is defined by the conditions x = sin y and –/2  y  /2. In words, arcsin x is the angle between –/2 and /2 whose sine is x.