Integration: antiderivatives, areas and accumulation
Integration in this paper is mainly computational and interpretive: find antiderivatives, apply Barrow's rule, compute the area between two curves, and explain the meaning of a definite integral of a marginal function.
Core rules
- Standard antiderivatives: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1, ∫(1/x)dx = ln|x| + C, ∫e^x dx = e^x + C; linearity lets you split sums and pull out constants.
- Substitution handles composites: with u = g(x), ∫f(g(x))g'(x)dx = ∫f(u)du.
- Barrow's rule: ∫_a^b f(x)dx = F(b) − F(a) for any antiderivative F of f.
- The definite integral is a signed area: parts below the axis count negatively, so a geometric area needs a sign study or a split at the zeros.
- Area between two curves: find the intersections a < b, decide which curve is above on each subinterval, and integrate upper minus lower; split where they cross.
- Accumulation: the integral of a marginal function gives the total change, e.g. ∫_0^q MC gives the variable cost of producing q units.
Worked example
Find the area enclosed by y = x² and y = 2x. Intersections: x² = 2x gives x = 0 and x = 2. On (0,2) the line is above the parabola, since 2x − x² = x(2 − x) > 0. So Area = ∫_0^2 (2x − x²)dx = [x² − x³/3]_0^2 = (4 − 8/3) − 0 = 4/3 square units. Had the question asked for ∫_0^2 x²dx the answer would be 8/3, so checking which function lies on top before subtracting is essential.
Traps that cost marks
- Omitting +C in an indefinite integral.
- Subtracting in the wrong order and reporting a negative area.
- Forgetting to split the integral where the curves cross or the function changes sign.
- Not solving f(x) = g(x) to obtain the limits of integration.
- Sign or arithmetic slips in F(b) − F(a); it is upper limit minus lower limit.