The area S enclosed by two curves y = f(x) and y = g(x) （f ≧ g）, together with x = a and x = b, is
S = ∫_a^b (f(x) − g(x)) dx
Example ① The area enclosed by y = x and y = x².
Solution:Intersection points: x = x² ⇒ x = 0, 1. On [0,1], x ≧ x².
S = ∫_0^1 (x − x²) dx = [x²/2 − x³/3]_0^1 = 1/2 − 1/3 = 1/6
Solution:Intersection points: x = x² ⇒ x = 0, 1. On [0,1], x ≧ x².
S = ∫_0^1 (x − x²) dx = [x²/2 − x³/3]_0^1 = 1/2 − 1/3 = 1/6
Key Point
- Integrate “the upper function − the lower function”
- Find the intersection points to determine the interval of integration
Practice Problems
- The area enclosed by y = x² and y = 2x
Answer and Explanation
- Answer:4/3
Intersection points x=0,2. ∫_0^2(2x−x²)dx = 4−8/3 = 4/3.