Area under and between curves

Area under a curve

• If y = f(x) is above the x-axis on [a, b], the area is A = ∫_a^b y dx.
• If the curve dips below the x-axis, integrate the absolute value, or split the interval and take the magnitude of the negative part.
• Example: the area under y = x² from x = 0 to x = 3 is ∫₀³ x² dx = [x³/3]₀³ = 9.

Area between two curves

• A = ∫_a^b (upper curve − lower curve) dx, where a and b are the x-coordinates of the intersection points.
• Example: the area between y = x and y = x². Intersections: x = x² → x = 0 or x = 1. On (0, 1), x > x², so A = ∫₀¹ (x − x²) dx = ½ − ⅓ = 1/6.
• Example: between y = 4 − x² and the x-axis. Intersections at x = ±2: A = ∫₋₂² (4 − x²) dx = [4x − x³/3]₋₂² = 32/3.

Choosing the variable

• If the region is easier to describe as x as a function of y, use A = ∫ (right curve − left curve) dy.
• Example: between x = y² and x = y + 2 → intersections y = −1, y = 2 → A = ∫₋₁² (y + 2 − y²) dy = 9/2.