Integration
1. Integration as reverse differentiation
Integration recovers a family of antiderivatives; boundary/point information selects one member.
Power rule
\[\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\qquad(n
e-1)\]Exponential
\[\int e^{ax}\,dx=\frac1a e^{ax}+C\]Trig
\[\int\sin ax\,dx=-\frac1a\cos ax+C,\qquad \int\cos ax\,dx=\frac1a\sin ax+C\]Find the curve from its derivative
If \(dy/dx=6x^2-4x+3\), then \[y=2x^3-2x^2+3x+C.\] Use a given point on the curve to determine \(C\).Common mistake. For an indefinite integral, include \(+C\). For a definite integral, the constant cancels and is not written.
2. Kinematics
Kinematics is repeated differentiation/integration plus careful interpretation of signs and constants.
Relationships
\[v=\frac{ds}{dt},\qquad a=\frac{dv}{dt}=\frac{d^2s}{dt^2}\]Reverse direction
\[v=\int a\,dt,\qquad s=\int v\,dt\]From acceleration to displacement
Given \(a=6t-4\), integrate: \[v=3t^2-4t+C_1.\] Use initial velocity to find \(C_1\), then integrate again: \[s=t^3-2t^2+C_1t+C_2,\] and use an initial position for \(C_2\).Exam tip. Velocity can be negative; speed is \(|v|\). Total distance may require splitting at times when \(v=0\).
3. Areas of finite regions
Accurate region identification is as important as integration.
Area to x-axis
\[A=\int_a^b y\,dx\quad\text{if }y\ge0\]Area between curves
\[A=\int_a^b(\text{upper}-\text{lower})\,dx\]- Find intersection points exactly; they become limits.
- Decide which curve is above on each interval.
- Split the integral if the order changes or a curve crosses the x-axis.
- Evaluate exactly first, then round only if requested.
Curve crosses x-axis
If \(f(x)\) changes sign at \(x=c\) inside \([a,b]\), geometrical area is \[A=-\int_a^c f(x)\,dx+\int_c^b f(x)\,dx\] when \(f<0\) then \(f>0\).Common mistake. A definite integral gives signed area. A question asking for “area” normally requires positive geometrical area.
4. Volumes of revolution
Volume questions combine geometry of circular cross-sections with definite integration.
About x-axis
\[V=\pi\int_a^b y^2\,dx\]About y-axis
\[V=\pi\int_c^d x^2\,dy\]Disc method
For \(y=\sqrt{x}\), rotated about the x-axis from \(x=1\) to \(x=4\): \[V=\pi\int_1^4(\sqrt{x})^2dx=\pi\int_1^4x\,dx=\frac{15\pi}{2}.\]Exam tip. Square the radius function before integrating. If rotating about the y-axis with a formula in x, you may need to rearrange for \(x\) in terms of \(y\).
5. Integration identities and exact proof
A clean exact chain earns method marks even before the final simplification.
Proof workflow
When asked to “show that” an area or volume equals an exact expression, keep fractions, surds and \(\pi\) exact throughout. Show the antiderivative, substitution of limits, and simplification explicitly.Exam tip. Do not replace exact values by calculator decimals in a proof question; decimal agreement is not an algebraic proof.