Integral calculus and applications to area and volume
Integral calculus and applications to area and volume
This shared calculus guide practises antiderivatives, definite integrals and geometric applications. It does not assert that any particular current PAU option must contain a specified integral question.
Core rules
- Barrow's rule: the definite integral from a to b of f is F(b) - F(a) for any antiderivative F. The constant of integration is irrelevant in the definite case.
- Techniques: substitution when the integrand contains g(x) times g'(x); integration by parts with the integral of u dv equal to uv minus the integral of v du, choosing u to become simpler when differentiated (ln x, inverse trigonometric functions, polynomials) and dv as the rest; partial fractions for rational functions, treating distinct linear, repeated linear and irreducible quadratic factors separately.
- Area between curves: find the intersection points, then integrate (upper curve minus lower curve) over each subinterval. If the curves cross inside the interval, split it, since the integral of f - g can cancel. Integrating with respect to y, use right minus left.
- Volume of revolution: about the x-axis, V = pi times the integral of f(x)^2; between two curves, pi times the integral of (f^2 - g^2); about the y-axis by shells, 2 pi times the integral of x f(x).
- Improper integrals are defined as limits of proper integrals and converge only when that limit is finite.
Worked example
Find the area enclosed by y = x^2 and y = x + 2. The intersections satisfy x^2 = x + 2, so x = -1 and x = 2; on that interval the line lies above the parabola. The area is the integral from -1 to 2 of (x + 2 - x^2) dx, whose antiderivative is x^2/2 + 2x - x^3/3. Evaluating gives 10/3 at x = 2 and -7/6 at x = -1, so the area is 10/3 + 7/6 = 9/2.
Traps that cost marks
Not locating the intersection points; integrating f - g across a crossing point and reporting a near-zero area; losing signs when evaluating at a negative limit; omitting the factor pi in a volume of revolution; mixing the disk and washer formulas when a gap exists between the curve and the axis; and treating an improper integral as if the endpoint could simply be substituted.