The curve intercepts the x axis at -3 and 3, and these values set the limits for integration. Regard the area under the curve as a set of very thin rectangles of width dx and length y where y=9-x^2. The area is the sum of the areas of the rectangles, the integral of ydx=(9-x^2)dx. The area is symmetrical about the y axis, so we can double the area between x=0 and +3:
[9x-x^3/3](3,0)=27-27/3=18, so the area is 2*18=36.