Integration: techniques, volumes of revolution, differential equations

Techniques of Integration 🔍

Think of integration like collecting all the tiny pieces of a puzzle to see the whole picture. Here are the main tools you’ll use:

1. Basic Rules – Power rule, Constant multiple rule, Sum rule.
   Example: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (neq -1)$$
2. Substitution – Like changing the scale of a picture to make it easier to see.
   Example: Let $u=x^2$, then $du=2x\,dx$ and $$\int x e^{x^2}\,dx = \frac12\int e^u\,du = \frac12 e^u + C = \frac12 e^{x^2} + C$$
3. Integration by Parts – Think of it as “take one part, leave the other.”
   Formula: $$\int u\,dv = uv - \int v\,du$$
4. Partial Fractions – Breaking a complicated fraction into simpler pieces, like splitting a chocolate bar into squares.
   Example: $$\int \frac{1}{x^2-1}\,dx = \int \frac{1}{(x-1)(x+1)}\,dx = \frac12\ln\left|\frac{x-1}{x+1}\right| + C$$
5. Trigonometric Substitution – When the integrand looks like a circle, use sine or cosine to simplify.
   Example: $$\int \sqrt{a^2-x^2}\,dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\!\left(\frac{x}{a}\right) + C$$

Exam Tip: Practice “Back‑Substitution”

After you finish an integral, always check if you can simplify the result by substituting back the original variable. It helps catch mistakes and often makes the answer look cleaner.

Volumes of Revolution 🎯

Imagine spinning a shape around an axis to create a 3‑D object. Two main methods:

1. Disc/Washer Method – Like slicing a cake into discs.
   $$V = \pi\int_a^b [R(x)]^2\,dx$$ where $R(x)$ is the distance from the axis to the curve. If there's a hole, subtract the inner radius: $$V = \pi\int_a^b \big([R_{\text{outer}}(x)]^2 - [R_{\text{inner}}(x)]^2\big)\,dx$$
2. – Think of wrapping a cylinder around the shape.
   $$V = 2\pi\int_a^b x\,f(x)\,dx$$ for rotation about the y‑axis, where $x$ is the radius of the shell and $f(x)$ its height.

Example: Find the volume of the solid obtained by rotating $y=\sqrt{x}$ from $x=0$ to $x=4$ about the x‑axis.

Solution (Disc Method):
$$V = \pi\int_0^4 (\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx = \pi\left[\frac{x^2}{2}\right]_0^4 = \pi\cdot8 = 8\pi$$

Exam Tip: Sketch the Region First

Before setting up the integral, draw the curve and the axis of rotation. It saves time and reduces errors.

Differential Equations 📐

A differential equation links a function with its derivatives. Two common types in P3:

1. First‑Order Linear – Looks like $$\frac{dy}{dx} + P(x)y = Q(x)$$. Solve by finding an integrating factor $μ(x)=e^{\int P(x)\,dx}$.
2. Separable – Can be written as $$\frac{dy}{dx} = f(x)g(y)$$. Separate variables: $$\frac{1}{g(y)}\,dy = f(x)\,dx$$ and integrate both sides.

Example (Separable): Solve $\frac{dy}{dx} = \frac{2x}{y}$ with $y(0)=1$.

Solution:
Separate: $$y\,dy = 2x\,dx$$
Integrate: $$\frac{y^2}{2} = x^2 + C$$
Use $y(0)=1$: $$\frac{1}{2} = 0 + C \Rightarrow C=\frac12$$
Final: $$y^2 = 2x^2 + 1 \Rightarrow y = \sqrt{2x^2 + 1}$$

Exam Tip: Check Units and Domain

When solving differential equations, remember that the solution must satisfy any given initial conditions and stay within the domain where the functions are defined.