# Fundamental theorem of calculus

Physics I · Calculus of a Single Variable · https://tryals.app/learn/physics-i/fundamental-theorem-of-calculus

## FTC Part 1 & The Bridge

The **Fundamental Theorem of Calculus** proves that differentiation and integration are **inverse operations**.

**Part 1 (Derivative of an integral):** If $F(x) = \int_{a}^{x} f(t)\,dt$, then:

$$F'(x) = f(x)$$

| Theorem Part | Operation | What It Does |
| --- | --- | --- |
| **Part 1** | Differentiation | Accumulation function derivative is $f$ |
| **Part 2** | Evaluation | Integrals use antiderivatives |

> **Common pitfall:** The two parts point in opposite directions. Do not mix up accumulation derivatives with antiderivative evaluation.

## FTC Part 2 & Techniques

**Part 2 (Evaluation):** If $F$ is **any** antiderivative of $f$ on $[a,b]$, then:

$$\int_{a}^{b} f(x)\,dx = F(b) - F(a)$$

**Core Techniques:**
- **Substitution:** Let $u=g(x)$, then $du=g'(x)\,dx$.
- **Integration by parts:** $\int u\,dv = uv - \int v\,du$.
- **Partial fractions:** Decompose rational integrands first.

**Improper integrals:** Used when limits are $\pm\infty$ or the integrand is unbounded.

$$\int_{1}^{\infty}\frac{1}{x^{2}}\,dx = \lim_{R\to\infty}\left[-\frac{1}{x}\right]_{1}^{R} = 1$$

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Match each FTC concept to its description.

**Answer:**

- FTC Part 1 → $d/dx[\int_a^x f(t)dt] = f(x)$
- FTC Part 2 → $\int_a^b f(x)dx$ = $F(b)$ − $F(a)$
- Antiderivative → Any F such that $F'(x) = f(x)$
- Net change theorem → $\int_a^b F'(x)dx$ = $F(b)$ − $F(a)$

**Why:** The net change theorem is FTC Part 2 read physically: total change = integral of rate.

Page: https://tryals.app/practice/physics-i/fundamental-theorem-of-calculus/match-each-ftc-concept-to-its-description

### 2. Which are direct consequences of the Fundamental Theorem of Calculus?

A. All bounded functions are integrable
B. Every continuous function on [a,b] has an antiderivative
C. Differentiation and integration are inverse operations
D. $\frac{d}{dx}\left[\int_a^x f(t)\,dt\right] = f(x)$ for continuous $f$

**Answer:** B. Every continuous function on [a,b] has an antiderivative; C. Differentiation and integration are inverse operations; D. $\frac{d}{dx}\left[\int_a^x f(t)\,dt\right] = f(x)$ for continuous $f$

**Why:** the first option overstates: not all bounded functions (e.g., Dirichlet function) are Riemann-integrable.

Page: https://tryals.app/practice/physics-i/fundamental-theorem-of-calculus/which-are-direct-consequences-of-the-fundamental-theorem-of-calculus

### 3. Order this chain of reasoning that explains **why** the Fundamental Theorem of Calculus works, for $A(x) = \int_a^x f(t)\,dt$.

**Answer:**

1. Define $A(x)$ as the accumulated area from $a$ to $x$
2. Nudge $x$ by a sliver $h$: the area gains a thin strip
3. That strip is nearly a rectangle: height $f(x)$, width $h$
4. So $\frac{A(x+h) - A(x)}{h} \approx f(x)$, exactly in the limit
5. Conclude $A'(x) = f(x)$: accumulation undoes differentiation

**Why:** The area function’s growth rate at $x$ is just the curve’s height there, one thin rectangle of insight. That is the bridge joining the two halves of calculus, and it is why antiderivatives compute areas.

Page: https://tryals.app/practice/physics-i/fundamental-theorem-of-calculus/order-this-chain-of-reasoning-that-explains-why-the-fundamental

### 4. The fundamental theorem links differentiation and integration as inverses. Why is that surprising given how the two are defined?

A. Because differentiation is exact and integration is approximate
B. Because the two operations are defined using the same limit and so must be inverse
C. Because integration was discovered many centuries before differentiation
D. Because one is local slopes and the other accumulated area

**Answer:** D. Because one is local slopes and the other accumulated area

**Why:** Slope and area have no obvious connection; the theorem asserts one. That is why it converts area problems into antiderivative problems and makes most integrals computable at all.

Page: https://tryals.app/practice/physics-i/fundamental-theorem-of-calculus/the-fundamental-theorem-links-differentiation-and-integration-as

### 5. FTC Part 1 states: if $F(x)$ = $\int_a^x f(t) dt$ then $F'(x) = f(x)$.

**Answer:** True

**Why:** FTC Part 1 is the statement that differentiation and integration are inverse operations.

Page: https://tryals.app/practice/physics-i/fundamental-theorem-of-calculus/ftc-part-1-states-if-f-x-a-x-f-t-dt-then-f-x-f-x