Area under a curve A=\int_a^b y\,dx Between the curve and the x-axis from x=a to x=b. Area between two curves A=\int_a^b\left(y_{\text{top}}-y_{\text{bottom}}\right)dx Upper curve minus lower curve. Find intersection points for the limits. ⚠
Areas below the axis
A definite integral counts area below the x-axis as negative. For a true geometric area that crosses the axis, split at the roots and add the absolute values. Worked example Definite integral Evaluate ∫₀² (3x² + 2) dx. Antiderivative: x³ + 2x. Evaluate at limits: [x³ + 2x]₀² = (8 + 4) − (0) = 12. ✓ The area is 12 square units.