Lecture ¹10

# Lecture ¹10. Improper Integrals Depending on a Parameter. Improper Multiple Integrals

Exercises for Lecture #10

The purpose of work:

The acquaintance with Improper Integrals Depending on a Parameter. Improper Multiple Integrals.

Theoretical information and examples

Content

2.8.1 Differentiation with respect to a parameter

2.8.2 Improper double and triple integrals

2.8.1 Differentiation with respect to a parameter. In the case of certain restrictions imposed on the functions f(x,a), f'(x,a) and on the corresponding improper integrals, we have Leibnitz's rule

Example 1. By differentiating with respect to a parameter, evaluate

Solution. If

then

whence

In order to find we set in the last equation and find

whence

2.8.2 Improper double and triple integrals

a) An infinite region. If a function f(x, y) is continuous in an unbounded region S, then we set

where s is a finite region lying entirely within S and signifies that we expand the region s by an arbitrary law so that any point of S should enter into it and remain in it. If there is a limit on the right hand side and if it does not depend on the choice of the region s, the corresponding improper integral is said to converge, otherwise it diverges.

If the integrand f(x, y) is non-negative then it is necessary and sufficient for the convergence of an improper integral that the limit on the right hand side of (1) exists at least for one system of regions s which exhaust the region S.

b) A discontinuous function. If a function f(x, y) is everywhere continuous in a bounded closed region S, except at the point P(a, b), then we set

Where is a region obtained from S by eliminating a small region of diameter which contains the point P. If (2) has a limit which does not depend on the type of small region eliminated from S, the improper integral under consideration is said to converge, otherwise to diverge.

If , then the limit on the right hand side of (2) does not depend on the type of regions eliminated from S; for instance, such regions may be circles of radius e/2 with centre at P.

The concept of improper double integrals is readily extended to the case of triple integrals.

Example 2. Test the convergence of

where S is the entire xy-plane.

Solution. Let be a circle of radius with centre at the origin. Passing to polar co-ordinates for, we have

If , then

and the integral diverges. However, if , then

and the integral converges. For , we have

i.e., the integral diverges.

Thus, Integral (3) converges for .