Fundamental Theorem of Calculus & area

Look back

Antiderivatives: substitution and $+C$ discipline carry straight into evaluating definite integrals with FTC.

Learning goals

  • State FTC parts I and II and connect “area with variable upper limit” to antiderivatives.
  • Evaluate definite integrals using antiderivatives: $\int_a^b f = F(b)-F(a)$.
  • Compute net signed area; interpret negative contribution below the $x$-axis.
  • Use average value of a function on $[a,b]$: $\dfrac{1}{b-a}\int_a^b f(x)\,dx$.

Concept sheet

FTC (evaluation form, Part II)

If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$, then $$ \int_a^b f(x)\,dx = F(b) - F(a). $$

FTC I (idea): If $g(x) = \int_a^x f(t)\,dt$ and $f$ is continuous, then $g'(x) = f(x)$. “Differentiation undoes integration with a fixed lower limit.”

Net area: Riemann sums → integral; regions below axis contribute negatively unless the problem asks for total geometric area (then split and add absolute areas).

Worked patterns

Pattern A — definite with substitution

Change limits with $u$: if $u = x^2+1$, when $x=0$, $u=1$; when $x=2$, $u=5$. Rewrite integral in $u$ and evaluate $F(5)-F(1)$.

Pattern B — area between curves

Between $x=a$ and $x=b$, if $f(x)\ge g(x)$, then area is $\int_a^b \big(f(x)-g(x)\big)\,dx$.

Pattern C — average value

Average of $f$ on $[a,b]$ is $\dfrac{1}{b-a}\int_a^b f(x)\,dx$. MVT for integrals guarantees some $c$ hits this height when $f$ is continuous.

Practice

Original drills — use Check when available, then Show solution. A quick celebration plays on correct auto-checks (respects reduced motion).

Homework / practice

Exam III on the syllabus covers §3.10 through §5.2. You have Review 3 below; chapter 5 section homework PDFs are still missing from materials/.

Section Link Focus
Review 3 (pack) Open PDF Before Exam III / final stretch
§5.1–§5.5 (placeholders) — Add PDFs for areas, Riemann sums, FTC, substitution when available

Mistakes & checks

  • Antiderivative $F$ must be valid on the full interval; watch discontinuities inside $[a,b]$.
  • Substitution: either change limits to $u$ or substitute back to $x$ before evaluating at endpoints.
  • “Area” wording vs “integral” — clarify signed vs geometric area in word problems.

Slide appendix (optional)

Optional condensed slide PDFs — see materials/README.md.

Quiz mode

Homework PDF quiz: each card links to the same WebAssign export filenames as the semester map (materials/). A session shows 20 shuffled cards from 45 PDF-linked slots; exact problem wording stays in the PDF unless you merge transcripts per quiz/transcripts/README.md. Regenerate: python3 scripts/gen_hw_quiz.py.