Calculus for Scientists and Engineers by Martin Brokate & Pammy Manchanda & Abul Hasan Siddiqi

Author:Martin Brokate & Pammy Manchanda & Abul Hasan Siddiqi
Language: eng
Format: epub
ISBN: 9789811384646
Publisher: Springer Singapore

(8.42)

The function f is then called integrable over . A formally precise definition of this limit will be given in Appendix D.10. It always exists when f is continuous, but it also exists for many discontinuous functions.

We now consider an arbitrary bounded region D of the plane. Let be an arbitrary rectangle which encloses D (that is, ), and let us denote by the function whose value is 1 for points in D and 0 at all other points. The integral of over is defined as

(8.43)

provided the integral on the right-hand side exists, that is, the product is integrable over Q. (Note that the function is discontinuous if , so that usually is discontinuous, too.) In particular, setting we obtain the area of D,

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