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LearnEdexcelYMA012018 specificationWMA14 · Pure Mathematics 4Integration

WMA14 · Module

Integration

7 subtopics≈ 300 min in total

Start the module: Integration by substitution Open graphing calculator

Integration by substitution
The reverse of the chain rule: replace part of the integrand by $u$ , change $dx$ into $du$ , change the limits, and turn a hard integral into an easy one.
≈ 50 min
Volumes of revolution
Spinning a region around an axis makes a solid. Slicing it into thin discs gives $V=\pi\displaystyle\int y^2\,dx$ about the $x$ -axis and $V=\pi\displaystyle\int x^2\,dy$ about the $y$ -axis, plus regions between two graphs.
≈ 45 min
Integration by parts
The reverse of the product rule: $\displaystyle\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$ . How to choose $u$ , when to use it twice, and the trick for $\int\ln x\,dx$ .
≈ 50 min
Forming differential equations
Turn words about change into mathematics. "Grows in proportion to", "decreases at a rate inversely proportional to", "flows in and leaks out": each becomes an equation for a derivative, ready to solve.
≈ 30 min
Separating the variables
Gather every $y$ with $dy$ on one side and every $x$ with $dx$ on the other, integrate, and add one constant. The method that solves almost every Paper 3 differential equation.
≈ 35 min
Particular solutions with harder integrals
Exam differential equations are integration questions in disguise. Once the variables are separated, one side often needs partial fractions, integration by parts or a trigonometric identity. Bring all of Paper 3 integration to bear, then pin down the constant.
≈ 45 min
Modelling with differential equations
Populations, cooling drinks, draining tanks and spreading rumours. Solve the equation, use two pieces of data to fix the constants, and then ask the real questions: when, how much, and what happens in the long run?
≈ 45 min