Symmetry can halve or quarter your computation. If the region is symmetric about the y-axis, compute the area on the right half and double it. If symmetric about both axes, compute the first-quadrant area and multiply by 4. Key examples: the ellipse $x^2$/$a^2$ + $y^2$/$b^2$ = 1 has four-fold symmetry, so total area = 4 * integral from 0 to a of bsqrt(1 - $x^2$/$a^2$) dx = pia*b. The region bounded by |x| + |y| = 1 has four-fold symmetry: first-quadrant area = integral from 0 to 1 of (1 - x) dx = 1/2, total = 4 * 1/2 = 2. Always sketch the region first and look for symmetry before integrating.