The Fundamental Theorem of Calculus (FTC) is the bridge between differentiation and integration — it's what makes calculus so powerful. The theorem has two parts: one tells us how to differentiate an integral, and the other tells us how to evaluate a definite integral using antiderivatives.
🎯 In this section you will learn
- FTC Part 1: $\frac{d}{dx} \int_a^x f(t) dt = f(x)$ — derivative of an integral
- FTC Part 2: $\int_a^b f(x) dx = F(b) - F(a)$ — evaluating definite integrals
- How to apply the FTC to compute areas and net change
📌 FTC Part 1: The Derivative of an Integral
$$ \frac{d}{dx} \int_a^x f(t) \, dt = f(x) $$
The derivative of the accumulation function is the original function
💡 Intuition
If $F(x) = \int_a^x f(t) dt$, then $F(x)$ represents the net area from $a$ to $x$. As $x$ increases a little, the area increases by approximately $f(x) \cdot dx$. So $F'(x) = f(x)$.
Find $\frac{d}{dx} \int_2^x t^3 dt$.
- By FTC Part 1, the derivative is just the integrand evaluated at the upper limit.
- $\frac{d}{dx} \int_2^x t^3 dt = x^3$
Find $\frac{d}{dx} \int_0^{x^2} \sin t \, dt$.
①
Apply FTC with Chain Rule
Let $u = x^2$. Then $\frac{d}{dx} \int_0^u \sin t \, dt = \sin(u) \cdot \frac{du}{dx}$
②
Simplify
$= \sin(x^2) \cdot 2x = 2x \sin(x^2)$
Find $\frac{d}{dx} \int_x^5 e^{t^2} dt$.
- $\int_x^5 e^{t^2} dt = -\int_5^x e^{t^2} dt$
- $\frac{d}{dx} \int_x^5 e^{t^2} dt = -\frac{d}{dx} \int_5^x e^{t^2} dt = -e^{x^2}$
📌 FTC Part 2: Evaluating Definite Integrals
$$ \int_a^b f(x) \, dx = F(b) - F(a) $$
where $F$ is any antiderivative of $f$ (i.e., $F'(x) = f(x)$)
💡 This changes everything!
Instead of computing limits of Riemann sums, we can find an antiderivative and evaluate it at the endpoints.
Find $\int_1^3 x^2 dx$.
①
Find an antiderivative
$F(x) = \frac{x^3}{3}$
②
Evaluate $F(b) - F(a)$
$F(3) - F(1) = \frac{27}{3} - \frac{1}{3} = 9 - \frac{1}{3} = \frac{26}{3}$
Find $\int_0^{\pi/2} \sin x \, dx$.
- Antiderivative of $\sin x$ is $-\cos x$
- $\int_0^{\pi/2} \sin x \, dx = [-\cos x]_0^{\pi/2} = (-\cos(\pi/2)) - (-\cos 0) = (0) - (-1) = 1$
Find $\int_0^1 e^x \, dx$.
- Antiderivative of $e^x$ is $e^x$
- $\int_0^1 e^x \, dx = [e^x]_0^1 = e^1 - e^0 = e - 1$
Find $\int_1^4 \frac{1}{x} \, dx$.
- Antiderivative of $\frac{1}{x}$ is $\ln|x|$
- $\int_1^4 \frac{1}{x} \, dx = [\ln|x|]_1^4 = \ln 4 - \ln 1 = \ln 4$
⚠️ Common Mistakes
- Forgetting the chain rule in FTC Part 1 when the upper limit is a function of $x$.
- Confusing order of evaluation: $F(b) - F(a)$, not $F(a) - F(b)$.
- Forgetting that definite integrals give a number — no $+C$ at the end!
- Using the wrong antiderivative: Always check that $F'(x) = f(x)$.
🔍 Key Takeaways
- FTC Part 1: $\frac{d}{dx} \int_a^x f(t) dt = f(x)$ — derivative of an integral.
- FTC Part 2: $\int_a^b f(x) dx = F(b) - F(a)$ — evaluate definite integrals using antiderivatives.
- FTC Part 2 is the workhorse for computing areas.
- When the upper limit is a function (e.g., $x^2$), use the Chain Rule with FTC Part 1.
- The FTC links differentiation and integration — the two main ideas of calculus.
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