Application of Definite Integration

For a continuous curve y =

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. STEP-BY-STEP: (1) Identify the cur

For curves y = f(x)

For curves y = f(x) and y = g(x), area between them is A = |∫[a to b] [f(x) - g(x)]dx|. CRITICAL STEPS: (1) Find intersection points by solving f(x) =

When f(x) ≤ 0 in [a

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|. Exa

Sometimes it's easier to integrate

Sometimes it's easier to integrate with respect to y instead of x. For curve x = g(y), area bounded by y-axis and lines y = c, y = d is A = ∫[c to d]

Use symmetry to simplify calculations

Use symmetry to simplify calculations: (1) If curve is symmetric about x-axis, calculate area in one half and multiply by 2, (2) If symmetric about y-