FTC Review The Method of Substitution February 4

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… 1. 5. 2. 6. 3. 7. 4.

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 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 1. Choose u. 2. Differentiate u w. r. t. x and solve for du. 3. Substitute u and du into the old integral involving x to form a new integral involving only u. 4. Antidifferentiate with respect to u. 5. 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) 1. Choose u = g(x). 2. Differentiate u w. r. t. x and solve for du. 3. 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). 4. 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 i) x = sin y and ii) – /2 y /2. In words, arcsin x is the angle between – /2 and /2 whose sine is x.