Application of Integrals
This short domain tests finding areas bounded by curves using definite integrals. Questions usually supply the curves and ask for the area of the region enclosed, sometimes requiring you to find the intersection points and sketch first.
Core rules
- Area under y = f(x) from x = a to x = b, above the x-axis, is ∫ₐ^b f(x) dx.
- Area between y = f(x) (upper) and y = g(x) (lower) from a to b is ∫ₐ^b [f(x) − g(x)] dx.
- If the curve crosses the x-axis inside the interval, split the integral and take absolute values, since signed area can cancel.
- Area with respect to the y-axis uses x = g(y) and limits on y: ∫ [right curve − left curve] dy.
- For a circle x² + y² = a², the area is πa², often obtained as 4 × ∫₀^a √(a² − x²) dx.
Worked example
Find the area between y = x and y = x² for 0 ≤ x ≤ 1. The curves meet where x = x², giving x = 0 and x = 1. Between these points x ≥ x², so the upper curve is y = x. Area = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6 square units.
Traps
- Not finding intersection points, so the limits are wrong.
- Subtracting the curves in the wrong order and reporting a negative area.
- Treating a region below the axis as positive without splitting the integral.
- Forgetting to multiply by the symmetry factor when using a quarter circle.