Area bounded by curves — A/L Combined Mathematics

What must I remember about Area bounded by curves?

• Area bounded by curve y = f(x), the x-axis, and vertical lines x = a and x = b: Area A = ∫_a^b |f(x)| dx; if f(x) ≥ 0, A = ∫_a^b f(x) dx; if f(x) ≤ 0, A = - ∫_a^b f(x) dx = |∫_a^b f(x) dx|.
• Area between two curves y = f(x) and y = g(x) from x = a to x = b: Area A = ∫_a^b [f(x) - g(x)] dx where f(x) ≥ g(x) on [a, b] (Upper curve minus Lower curve).
• Integration with respect to y: For regions bounded by x = f(y), the y-axis, and y = c, y = d: Area A = ∫_c^d f(y) dy; between two curves x1 = f(y) and x2 = g(y): Area = ∫_c^d [f(y) - g(y)] dy (Right curve minus Left curve).
• Curves intersecting at multiple points: Subdivide the interval [a, b] at each intersection point c where f(c) = g(c), integrating [f(x) - g(x)] where f ≥ g and [g(x) - f(x)] where g ≥ f.
• Volume of solid of revolution (Cartesian): Rotation of y = f(x) about x-axis between x = a and x = b: V = π ∫_a^b [f(x)]² dx; rotation about y-axis between y = c and y = d: V = π ∫_c^d [g(y)]² dy.

Test yourself (1 minute)

1. Sign trap in definite area integrals

   Never compute ∫_a^b f(x) dx directly across an x-intercept where f(x) changes sign; positive and negative regions cancel algebraically, understating physical geometric area.

2. Points of intersection as integration limits

   Set f(x) = g(x) to solve for intersection coordinates; always sketch the curves to identify which curve lies above the other.

3. Symmetry simplification

   If the region is symmetric across an axis (e.g., ellipse x²/a² + y²/b² = 1), compute area of one quadrant and multiply by 4: Total Area = 4 * ∫₀^a (b/a)√(a² - x²) dx = πab.

Practise Area bounded by curves questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 15

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Where does Area bounded by curves fit in the syllabus?

Year 1 · Term 2