Substitution, parts and volumes of revolution

What you’ll learn

• Select substitution, parts or partial fractions for an integral.
• Calculate parametric areas.
• Find volumes obtained by rotation about the x-axis.

Understand the idea

Substitution changes the variable to simplify the structure of an integrand. Replace the differential as well as every occurrence of the original variable. For a definite integral, either change both limits to the new variable or convert the antiderivative back before using the old limits.

Integration by parts reverses the product rule. Choose the differentiated factor so that it becomes simpler, while the other factor can be integrated. It is particularly useful for products involving polynomials, exponentials or logarithms; sometimes it must be repeated. Partial fractions instead break rational functions into terms with standard antiderivatives.

Rotation about the x-axis turns a thin vertical strip into a disk of area $\pi y^2$. Sum the disks using $V=\pi\int y^2\,dx$. For parametric curves, replace $dx$ by $(dx/dt)dt$, including in the area formula. Check the direction of the parameter limits so the geometric quantity is positive.

Key formulae

$$\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$$

$$u=g(x)\Rightarrow du=g'(x)\,dx$$

$$V=\pi\int_a^b y^2\,dx$$

$$\int y\,dx=\int y(t)x'(t)\,dt,\qquad V=\pi\int y(t)^2x'(t)\,dt$$

Worked example

Evaluate $\int_0^1 xe^x\,dx$.

1. Choose $u=x$ and $dv/dx=e^x$, so $du/dx=1$ and $v=e^x$.
2. By parts, $\int xe^x\,dx=xe^x-\int e^x\,dx=(x-1)e^x+C$.
3. Evaluate at the limits: $(1-1)e^1-(0-1)e^0=1$.

Answer: $1$.