Area from thin strips

Formulas from geometry measure figures with straight edges and circles. A region with a curved boundary needs integral calculus.

Take the region bounded by y = f(x) (with f ≥ 0), the x-axis and the vertical lines x = a and x = b. Slice it into thin vertical strips. A strip at position x has height y = f(x) and width dx, so its area is dA = y dx.

Adding all the strips from x = a to x = b and letting their width shrink gives the area: A = ∫ₐᵇ y dx = ∫ₐᵇ f(x) dx.

The strip dA = y dx is called the elementary area. It sits at an arbitrary x between a and b and stands for every strip at once.

Worked example: the region under y = x² from x = 0 to x = 3 has area ∫₀³ x² dx = [x³/3] from 0 to 3 = 9.

Worked example: the region under y = √x from x = 0 to x = 4 has area ∫₀⁴ x^(1/2) dx = (2/3) × 4^(3/2) = 16/3.

Worked example: under y = x² + 1 between x = 1 and x = 2 the area is [x³/3 + x] from 1 to 2 = 14/3 − 4/3 = 10/3.

With a finite number of strips the sum only approximates the area; more and thinner strips bring it closer. The integral is the exact value those sums approach.