TMUA Integration with the Power Rule

The TMUA syllabus restricts integration to the power rule on polynomials and negative or fractional powers — no exponentials, no logarithms, no substitution, no integration by parts. Nearly every Paper 1 has one or two questions in this territory, and the standard application is the area enclosed between two curves.

• Direct integration. $\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \ne -1$.
• Definite integration. Newton–Leibniz: $\int_a^b f = F(b) - F(a)$. Limits matter; swapping them flips the sign.
• Area between curves. Find the abscissae where the curves meet, then integrate (upper minus lower) between those abscissae.
• Recovering a function from its derivative. Integrate; a single point’s value fixes the constant.

The move. Area enclosed between two curves equals the definite integral of upper minus lower between the intersection abscissae. Find the intersections first, identify which curve is on top, then integrate the difference.