The Fundamental Theorem of Calculus
The theorem that ties the whole course together: differentiation and integration are inverse operations. It is what lets us compute integrals without ever returning to sums.
- State both parts of the FTC.
- Explain why the area function's derivative is $f$.
- Evaluate definite integrals with antiderivatives.
- Differentiate integrals with variable limits.
It would be astonishing if two operations invented for unrelated reasons — slopes and areas — turned out to be exact inverses. They do. That is the Fundamental Theorem, and it is why integration becomes "find an antiderivative" instead of summing forever.
Because of it we can solve the differential equations that model nearly everything — orbits, circuits, epidemics (SIR models), and option prices (Black–Scholes). Without it those would be intractable.
Newton's own teacher, Isaac Barrow, glimpsed the connection geometrically; Newton and Leibniz saw it in full. This is the theorem that turned calculus from a collection of tricks into the central tool of science. The name is not an exaggeration.
The two directions of the theorem
FTC Part 1 says integration creates an antiderivative. FTC Part 2 says an antiderivative evaluates an integral. Keeping those directions separate prevents most mistakes in this unit.
Variable-limit checklist
- Upper limit $g(x)$ contributes $f(g(x))g'(x)$.
- Lower limit $h(x)$ contributes $-f(h(x))h'(x)$.
- The integration variable is a dummy variable; it should not be confused with the outside variable $x$.
- If the integrand also contains $x$ outside the dummy variable, the simple FTC rule may not apply directly.
Part 1 — the area function
If $f$ is continuous on $[a,b]$ and $F(x)=\displaystyle\int_a^x f(t)\,dt$, then $F$ is differentiable on $(a,b)$ and $F'(x)=f(x)$. At the endpoints, the corresponding one-sided derivative has the same value.
Part 2 — evaluation
If $f$ is continuous and $G$ is any antiderivative of $f$ (so $G'=f$), then $$\int_a^b f(x)\,dx=G(b)-G(a).$$
$\displaystyle\int_1^e\frac{\ln x}{x}\,dx$. With $u=\ln x$ an antiderivative is $\tfrac12(\ln x)^2$, so the value is $\tfrac12(1-0)=\tfrac12$.
Variable limits and the chain rule
$\dfrac{d}{dx}\displaystyle\int_{a}^{g(x)}f(t)\,dt=f\!\big(g(x)\big)\,g'(x).$ With both limits variable, differentiate each and subtract.
$\dfrac{d}{dx}\displaystyle\int_{0}^{x^2}\cos t\,dt=\cos(x^2)\cdot2x.$
• In Part 1 the variable is the upper limit; the dummy variable
inside is different ($t$, not $x$).
• With a variable upper limit $g(x)$, don't forget the $g'(x)$ factor.
• Part 2 needs $f$ continuous on $[a,b]$ and an antiderivative $G$ with
$G'=f$ throughout $(a,b)$.
Practice
Problems adapted from the MAT137 question bank by Alfonso Gracia-Saz, used under CC BY-SA 4.0. Write full solutions before opening the answer.
Compute $\dfrac{d}{dx}\displaystyle\int_{x}^{3}e^{-t^2}\,dt$.
Answer
Evaluate $\displaystyle\int_0^{\pi}\sin x\,dx$.
Answer
Find $F'(x)$ if $F(x)=\displaystyle\int_{1}^{x^3}\frac{dt}{1+t^2}$.
Answer
If $\displaystyle\int_0^x f(t)\,dt=x\,e^{x}$, find $f$.
Answer
Find $F'(x)$ for $F(x)=\displaystyle\int_{2x}^{x^2}\frac{dt}{1+t^2}$.
Answer
Evaluate $\displaystyle\int_{\pi/4}^{\pi/3}\sec^2x\,dx$.
Answer
Evaluate $\displaystyle\int_0^1\!\left[e^x+e^{-x}-\cos(2x)\right]dx$.
Answer
Suppose $f$ is continuous and $\displaystyle\int_0^x e^t f(t)\,dt =\frac{\sin x}{x^2+1}$ for all $x$. Find $f(x)$.
Solution
Expanded topic checklist
Course coverage. This unit is part of the complete MAT137 topic map. It should cover the following items before you move on:
- FTC Part 1. Differentiate accumulation functions and handle variable upper or lower limits.
- FTC Part 2. Evaluate definite integrals using antiderivatives under the right continuity hypotheses.
- Net change. Interpret integrals of rates as total change.
- Substitution connection. See substitution for definite integrals as a consequence of the chain rule and FTC.
Use the Alfonso Gracia-Saz MAT137 materials as the primary local spine, with the current UofT single-variable calculus topic split as the completeness check. The added items are original study prompts, not copied textbook exercises.
The spine is: accumulation creates antiderivatives, antiderivatives evaluate accumulation, and the two halves close the loop between differentiation and integration.
Deep practice extensions
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Differentiate $F(x)=\int_2^{x^3}\sqrt{1+t^4}\,dt$.
Guide
Differentiate $G(x)=\int_x^1 e^{-t^2}\,dt$.
Guide
If $v(t)=3t^2$ meters/sec, find displacement from $t=1$ to $t=3$.
Guide
Why does FTC Part 1 require continuity in the usual statement?