Integration and Its Applications
What the exam tests
Integration and Its Applications tests antiderivatives, the fundamental theorem, substitution, areas between curves, and volumes of revolution. Kyoto problems often ask for a region whose boundaries change, so splitting the integral is part of the work.
Core rules
- An indefinite integral is a family: ∫f(x) dx = F(x) + C, where F' = f.
- The fundamental theorem: ∫_a^b f(x) dx = F(b) - F(a) for a continuous f with antiderivative F.
- For substitution, change the limits when using a definite integral, or substitute back before evaluating.
- Area between curves: integrate (upper - lower) over the interval where that order holds. If the curves cross, split at the intersection points.
- Volume of revolution about the x-axis: V = π∫_a^b [f(x)]^2 dx, provided the region is correctly described.
Worked example
Find the area between y = x and y = x^2 for 0 ≤ x ≤ 1. The curves meet where x = x^2, so x = 0 or 1. On (0, 1), x > x^2, so the area is ∫_0^1 (x - x^2) dx = [x^2/2 - x^3/3]_0^1 = 1/2 - 1/3 = 1/6.
Traps
Do not omit the constant C in an indefinite integral. A definite integral can be negative even when it represents an area; use absolute value or the correct upper-minus-lower order for geometric area. When substituting, do not mix old limits with the new variable. Check whether the volume formula needs π and whether the axis of rotation matches the squared radius.