Vertical Strips (dx) vs Horizontal Strips (dy):

• Vertical strips work best when curves are given as y = f(x) and don't require splitting
• Horizontal strips work best for curves like $y^2$ = 4ax or x = g(y), and when vertical strips would require multiple integrals
• Example: area between x = $y^2$ and x = 2-$y^2$ needs one integral with dy but two with dx

Direct Integration vs Symmetry:

• Direct integration is always valid but may involve tedious computation
• Symmetry (even/odd/periodic) can reduce work by half or more
• Example: ellipse area with 4-fold symmetry needs only the first quadrant integral

Cartesian vs Parametric vs Polar:

• Cartesian is the default for most JEE problems
• Parametric is needed for curves like cycloids or when x and y are given in terms of a parameter t
• Polar is needed for cardioids, rose curves, and spirals

Signed Integral vs Geometric Area:

• Signed integral: direct output of the definite integral, can be zero or negative
• Geometric area: always positive, requires splitting at zeros and taking absolute values
• The problem statement determines which to compute: "area bounded by" means geometric area

Formula Shortcuts vs Full Integration:

• Standard results (8$a^2$/3 for parabola, |a|(beta-alpha)^3/6 for quadratic) save time
• Full integration is needed when standard results don't apply
• In JEE, using memorized formulas can save 2-3 minutes per problem