Question 85.

5.8 ADDITIONAL PROPERTIES OF THE DEFINITE INTEGRAL _ 281 For Exercises 85 and 86 reverse the roles of x and u in (5.7.2) quadrant is the region below the curve y = Vr2 - x2, x E and write [0, r]. Therefore A =4 72 _ x2 dx. 85. (The area of a circular region) The circle x? + y? = r2 en- closes a circular disc of radius r. Justify the familiar formula Set x = r sinu, dx = r cosu du. A = nr2 by integration. HINT: The quarter-disk in the first 86. Find the area enclosed by the ellipse b2x2 + a2y2 = a262. 5.8 ADDITIONAL PROPERTIES OF THE DEFINITE INTEGRAL We come now to some properties of the definite integral that we'll make use of time and time again. Some of the properties are pretty obvious; some are not. All are important. I. The integral of a nonnegative continuous function is nonnegative: (5.8.1) if f(x) > 0 for all x E [a, b], then (' f( )dx 2 0