Barrons AP Calculus - David Bock

The limit on the right in the starred equation is, by definition, the derivative of F(x), that is, f (x).

EXAMPLE 20

Reexpress in terms of u if

SOLUTION: When u^2 = x − 2, and 2u du = dx. The limits of the given integral are

values of x. When we write the new integral in terms of the variable u, then the limits, if written,

must be the values of u that correspond to the given limits. Thus, when x = 3, u = 1, and when x =

6, u = 2. Then

EXAMPLE 21

If g ′ is continuous, find

SOLUTION:

Note that the expanded limit is, by definition, the derivative of g(x) at c.

C. INTEGRALS INVOLVING PARAMETRICALLY DEFINED

FUNCTIONS

The techniques are illustrated in Examples 22 and 23.

BC ONLY

EXAMPLE 22

Evaluate where x = 2 sin θ and y = 2 cos θ.

SOLUTION: Note that dx = 2 cos θ dθ, that when x = −2, and that when x = 2.