University Mathematics Handbook (2015)

VIII. Integral Calculus of Multivariable Functions

Chapter 1. Parameter-Dependent Integral

1.1  Definition

Let be function defined on rectangular

a.  If is integrable by in interval , for every fixed of , then function is integral dependent of the parameter.

b.  If is integrable by on for every , then function is integral dependent of the parameter.

1.2  Properties of Parameter-Dependent Integral

a.  If function is continuous on rectangle then is continuous on and is continuous on .

b.  Leibniz Rule

1.  If and are continuous on , then

2.  If and are continuous on , then

c.  If is continuous on , then

d.  If is continuous on and functions , are continuous on , then is continuous on .

e.  If and are continuous on , and functions , are derivable, then function is derivable, and