Applications of integrals

The essentials

If y = f(x) is continuous and non-negative on [a, b], the area of the region bounded by the curve, the x-axis and the lines x = a and x = b is ∫ₐᵇ y dx. You can think of it as the sum of thin vertical strips of height y and width dx. Similarly, the area between a curve x = g(y), the y-axis and the lines y = c and y = d is ∫ from c to d of x dy, using horizontal strips.

If the curve lies below the x-axis, the integral is negative, so take its absolute value for the area. If the curve crosses the axis, split the integral at the crossing points.

Key points

• Always sketch the region before writing the integral.
• Use symmetry: the circle x² + y² = a² has area 4 × (area in the first quadrant) = πa².
• First-quadrant area of x² + y² = a² is ∫₀ᵃ √(a² − x²) dx = πa²/4.
• Ellipse x²/a² + y²/b² = 1 has area πab.
• Parabola y² = 4ax is symmetric about the x-axis; the area enclosed with the line x = c is 2∫₀ᶜ 2√(ax) dx.
• ∫ √(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C.

Exam tips

• Identify where the region starts and ends (the limits) from the sketch, not by guessing.
• For symmetric figures, compute one part and multiply.

Common mistakes

• Reporting a negative number as an area.
• Forgetting to double the integral for a region symmetric about the x-axis, such as inside a parabola y² = 4ax.