# The Fundamental Theorem of Calculus

Mathematics I · Integral Calculus · https://tryals.app/learn/mathematics-i/the-fundamental-theorem-of-calculus

## Uniting Calculus

Integration has worn two faces: **antiderivatives** (reversing differentiation) and **definite integrals** (accumulated area). The **Fundamental Theorem of Calculus (FTC)** reveals they are two sides of one coin.

| Part | Formula / Statement | Meaning | --- | --- | --- |
| **Part 1** | $F(x) = \int_a^x f(t)\, dt \implies F'(x) = f(x)$ | Derivative of accumulated area is curve height | 
| **Part 2** | $\int_a^b f(x)\, dx = F(b) - F(a)$ | Practical evaluation rule via antiderivatives |

Conceptually, accumulation and rate-of-change are inverse operations. This is why velocity integrates to displacement and any rate integrates to a total.

## The Evaluation Rule

Part 2 is the practical powerhouse. You never need Riemann sums for exact integrals; just find an antiderivative and subtract endpoint values.

**Worked Example:** Evaluate $\int_0^2 x\, dx$.
1. Find antiderivative: $F(x) = \frac{x^2}{2}$.
2. Subtract limits: $\left[\frac{x^2}{2}\right]_0^2 = \frac{2^2}{2} - \frac{0^2}{2} = 2 - 0 = 2$.

**Common Pitfall:** Subtracting endpoints in the **wrong order**. The rule is **$F(b) - F(a)$** (upper minus lower); reversing it flips the sign.

Note that **any** antiderivative works. The $+C$ always cancels out during subtraction $F(b) - F(a)$.

## Practice questions

3 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. The Fundamental Theorem shows that differentiation and integration are inverse operations.

**Answer:** True

**Why:** True. Part 1 says differentiating the accumulated-area function returns the integrand; Part 2 says a definite integral can be evaluated by any antiderivative. Linking area to slope, two apparently unrelated ideas, is what turned integration from a limit computation into an algebraic one.

Page: https://tryals.app/practice/mathematics-i/the-fundamental-theorem-of-calculus/the-fundamental-theorem-shows-that-differentiation-and-integration

### 2. Order the steps for evaluating a definite integral with Part 2 of the Fundamental Theorem.

**Answer:**

1. Check that the integrand is continuous on the interval from a to b
2. Find any one antiderivative F of the integrand
3. Evaluate F at the upper limit b
4. Evaluate F at the lower limit a
5. Subtract to obtain F(b) - F(a)

**Why:** Continuity licenses the theorem; then any antiderivative will do, because the + C cancels in the subtraction. That last fact is why the recipe never asks you which antiderivative to pick.

Page: https://tryals.app/practice/mathematics-i/the-fundamental-theorem-of-calculus/order-the-steps-for-evaluating-a-definite-integral-with-part-2-of-the

### 3. The Fundamental Theorem lets you evaluate an integral by finding any antiderivative. Why does *any* one work, when there are infinitely many?

A. Because the constant is always zero for definite integrals
B. Because the theorem specifies which antiderivative to use
C. Because only one antiderivative is valid on the interval of integration
D. Because they differ only by a constant, which cancels in F(b) − F(a)

**Answer:** D. Because they differ only by a constant, which cancels in F(b) − F(a)

**Why:** The +7 appears at both ends and subtracts away, so every antiderivative gives the same difference. That is what makes the theorem usable, you may take the laziest antiderivative you can find, and the ambiguity that makes indefinite integration a family is exactly the part the definite integral discards.

Page: https://tryals.app/practice/mathematics-i/the-fundamental-theorem-of-calculus/the-fundamental-theorem-lets-you-evaluate-an-integral-by-finding-any