Applications of Integrals
Applications of Integrals
The exam tests the area under a curve, the area between two curves, and standard areas such as those of circles, ellipses and parabolas using definite integrals.
Core rules
- Area under y = f(x) from x = a to x = b is ∫ₐᵇ |f(x)| dx. Split the interval where the curve crosses the x-axis.
- Area between y = f(x) (upper) and y = g(x) (lower) from a to b is ∫ₐᵇ [f(x) − g(x)] dx.
- Find intersection points first; they usually give the limits of integration.
- For x = g(y), use horizontal strips: area = ∫ [right curve − left curve] dy.
- Use symmetry to double an area when the region is mirror-symmetric about an axis.
Worked example
Find the area between y = x² and y = x.
Intersections: x² = x gives x = 0 and x = 1. On (0,1), x > x², so x is the upper curve. Area = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6 square units.
Traps
- Subtracting in the wrong order and reporting a negative area.
- Forgetting to split at an x-intercept when part of the region lies below the axis.
- Using x-limits for a region more naturally described by horizontal strips.
- Missing intersection points, so the limits are wrong.
- Rushing the sketch; an accurate sketch prevents most upper/lower errors.