The Fundamental Theorem of Calculus

Part 2, Chapter 2: Single-Variable Real Analysis

Learning objectives

  • State both parts of the Fundamental Theorem of Calculus (FTC1 and FTC2)
  • Use FTC1 to differentiate accumulation functions, possibly composed with the chain rule
  • Use FTC2 to evaluate definite integrals via antiderivatives
  • Derive integration by parts and the change-of-variables formula from FTC2

The Fundamental Theorem of Calculus is the single most important theorem in calculus. It says two seemingly unrelated operations, differentiation (rates of change) and integration (accumulated area), are inverses of each other. Before this connection was made rigorous (Cauchy and Riemann, 1800s), the practical "trick" of computing integrals by guessing antiderivatives was an empirical shortcut without theoretical justification. After FTC, it became the centerpiece of the subject: every integration technique, integration by parts, substitution, partial fractions, ultimately derives from one of FTC's two parts.

FTC Part 1: differentiating the accumulation function

If ff is continuous on [a,b] and we define the accumulation function F(x)=intaxf(t),dtF(x)=\int_a^x f(t)\,dtaxf(t),dt, then F′(x)=f(x)F'(x)=f(x) on (a,b)(a,b). In words: the derivative of the area-from-aa-to-xx function is the original function. This guarantees that every continuous function has an antiderivative (a function whose derivative is ff), namely the accumulation function itself.

FTC Part 2: evaluating definite integrals

If ff is continuous on [a,b] and FF is ANY antiderivative of ff (meaning F′=fF'=f), then intabf(x),dx=F(b)−F(a)\int_a^b f(x)\,dx=F(b)-F(a)abf(x),dx=F(b)−F(a). This is the computational powerhouse: instead of computing Riemann sums or chasing supL=infU\sup L=\inf U, we find an antiderivative and evaluate at the endpoints. The proof of Part 2 uses MVT applied to each subinterval of a partition, another payoff of §2.3.

FTC1 with the chain rule

If G(x)=intau(x)f(t),dtG(x)=\int_a^{u(x)}f(t)\,dtau(x)f(t),dt for a differentiable uu, then by the chain rule G′(x)=f(u(x))cdotu′(x)G'(x)=f(u(x))\cdot u'(x). Example: for G(x)=int0x2sin(t),dtG(x)=\int_0^{x^2}\sin(t)\,dt0x2sin(t),dt, G′(x)=sin(x2)cdot2xG'(x)=\sin(x^2)\cdot 2x. This is the most common "trick" question in early analysis: students forget the chain-rule factor and write just sin(x2)\sin(x^2).

Plot F(x)=int0xt2,dt=x3/3F(x)=\int_0^x t^2\,dt = x^3/30xt2,dt=x3/3 and its derivative f(x)=x2f(x)=x^2. The graphs visually demonstrate FTC1: at every xx, the slope of FF equals f(x)f(x). You can also see FTC2 in action by computing F(2)−F(0)=8/3F(2)-F(0)=8/3, which should equal int02x2,dx\int_0^2 x^2\,dx02x2,dx (true integral 8/3 by hand or by Riemann sums).

Where this shows up
  • Physics & conservation laws: work-energy theorem says W=intFcdotdx=DeltaKEW=\int F\cdot dx=\Delta KE. The line integral of force equals the change in kinetic energy, a direct application of FTC2 with F=ma=m,dv/dtF=ma=m\,dv/dt.
  • Probability cumulative distributions: a continuous random variable has density f(x)f(x) and CDF F(x)=int−inftyxf(t),dtF(x)=\int_{-\infty}^x f(t)\,dt−inftyxf(t),dt. FTC1 says the derivative of the CDF is the density: F′(x)=f(x)F'(x)=f(x). Every density estimator and Monte Carlo simulation is built on this relationship.
  • Differential equations & integral equations: the solution to y′=g(x),y(a)=y0y'=g(x), y(a)=y_0 is y(x)=y0+intaxg(t),dty(x)=y_0+\int_a^x g(t)\,dtaxg(t),dt, a literal FTC1 statement. Numerical ODE solvers (Euler, Runge-Kutta) approximate this integral.

    Pause and think: FTC2 says intabf=F(b)−F(a)\int_a^b f=F(b)-F(a)abf=F(b)−F(a) for ANY antiderivative FF. Two different antiderivatives differ by a constant. Why does the choice of constant not change the answer?

    Try it

    • Predict first: compute int02(4x3−2x),dx\int_0^2 (4x^3-2x)\,dx02(4x3−2x),dx using FTC2. The antiderivative is x4−x2x^4-x^2; evaluate at the endpoints.
    • Let G(x)=intsinxx2et,dtG(x)=\int_{\sin x}^{x^2} e^t\,dtsinxx2et,dt. Find G′(x)G'(x) using FTC1 plus chain rule for BOTH the upper and lower limits.
    • Prove integration by parts \int_a^b u\,dv = [uv]_a^b - \int_a^b v\,du from the product rule plus FTC2. (Integrate (uv)′=u′v+uv′(uv)'=u'v+uv' from aa to bb.)
    • True or false: if int0xf(t),dt=0\int_0^x f(t)\,dt=00xf(t),dt=0 for every xx, then ff is identically zero on its domain of continuity. Justify using FTC1.

      A trap to watch for

      FTC1 requires ff continuous on [a,b]. If ff has a discontinuity (even just one finite jump), F(x)=intaxfF(x)=\int_a^x faxf is still continuous but it is NOT differentiable at the jump, FTC1 only gives F′(x)=f(x)F'(x)=f(x) where ff is continuous. For Lebesgue integration, even more subtle: FF is absolutely continuous and differentiable almost everywhere, but the relationship is via the Lebesgue Differentiation Theorem, not Riemann's FTC.

      What you now know

      You can apply FTC1 and FTC2 confidently, including the chain-rule variant; derive integration by parts and substitution from FTC2; and connect each piece to its real-world cousin in physics, probability, and ODEs. The next two sections (§2.6 Pointwise convergence and §2.7 Uniform convergence) ask when limntoinftyintfn=intlimntoinftyfn\lim_{n\to\infty}\int f_n=\int\lim_{n\to\infty}f_nn=intlimntoinftyfn, a question that turns out to be subtle and requires the new notion of uniform convergence.

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      References

      • Garrity, T. (2002). All the Mathematics You Missed. Cambridge University Press, ch. 2.
      • Spivak, M. (2008). Calculus (4th ed.). Publish or Perish, ch. 14.
      • Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley, ch. 7.
      • Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill, ch. 6.
      • Bartle, R. G., Sherbert, D. R. (2011). Introduction to Real Analysis (4th ed.). Wiley, ch. 7.

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