Index Numbers — Formula Sheet

The definite integral ∫ᵇₐ f(x) dx gives the area under the curve y = f(x) from x = a to x = b

If f(x) ≥ 0 in [a,b], then area = ∫ᵇₐ f(x) dx

If f(x) ≤ 0 in [a,b], then area = |∫ᵇₐ f(x) dx|

Area between curves y = f(x) and y = g(x) is |∫ᵇₐ [f(x) - g(x)] dx|

Parabolas

y² = 4ax, area with latus rectum = 32a²/3

Circles

x² + y² = r², total area = πr²

Ellipses

x²/a² + y²/b² = 1, total area = πab

For a continuous curve y = f(x) where f(x) ≥ 0 in interval [a,b], the area under the curve is A = ∫[a to b] f(x)dx.

For curves y = f(x) and y = g(x), area between them is A = |∫[a to b] [f(x) - g(x)]dx|.

When f(x) ≤ 0 in [a,b], the definite integral gives negative value, but area is always positive.

Solution

Take absolute value |∫[a to b] f(x)dx|.