Advanced Calculus fi..

228 Advanced Calculus, Fifth EditionTHEOREM If f (x, y) is continuous in a closed region R described by the inequalities(4.31), then, for xl i x 5 x2,is a continuous function of x, andSimilarly, if R is described by (4.32),;d4For a proof of existence of the double integral of a continuous function and ofthe reduction to an iterated integral, the reader is referred to Section 4.11.EXAMPLE Let R be the quarter-circle, 0 5 y 5 1/-, 0 5 x 5 1, and letf (x, y) = x2 + y2. Then one hasThe double integral can be interpreted and applied in a variety of ways. Thefollowing are illustrations:I. Volume. If z = f (x, y) is the equation of a surface, thengives the volume between the surface and the xy-plane, volumes above thexy-plane being counted positively and those below thexy-plane being countednegatively.11. Area. If one takes f (x, y) E 1, one obtains