Integrated single-variable calculus problems

This lesson combines all tools — limits, derivatives, integrals, Taylor series, and complex numbers — into multi-step problem solving.

Synthesis checklist

1. Model the system — Translate the physical description into a function $f(x)$ with a clear domain.
2. Differentiate — Find critical points, rates of change, or slopes.
3. Integrate — Compute totals (area, work, accumulated quantity).
4. Approximate — Use Taylor polynomials when exact solutions are impractical.
5. Validate — Check units, signs, boundary behaviour, and limiting cases.

Example: area between curves

$$A = \int_{a}^{b}\bigl|f(x)-g(x)\bigr|\,dx$$

Example: arc length

$$L = \int_{a}^{b}\sqrt{1+\bigl(f'(x)\bigr)^{2}}\,dx$$

Complex-number toolkit

• Modulus: $|z| = \sqrt{a^{2}+b^{2}}$.
• Conjugate: $\overline{z} = a - bi$, so $z\,\overline{z}=|z|^{2}$.
• Roots of unity: The $n$-th roots of 1 are $e^{2\pi i k/n}$, $k=0,\ldots,n-1$.

Strategy: When facing a multi-step problem, sketch the function first. A good graph reveals domain restrictions, symmetry, and the location of extrema — all before you compute a single derivative.

Common pitfall: In applied problems the hardest step is not the calculus but the setup: naming variables, writing the constraint, identifying what is being extremized or accumulated. If the setup is wrong, perfect differentiation cannot save it.