# The Fundamental Theorem of Calculus

Computer Science I · Calculus · https://tryals.app/learn/computer-science-i/the-fundamental-theorem-of-calculus

## Two Faces Become One

Integration has worn two faces: **antiderivatives** (reversing differentiation) and **definite integrals** (accumulated area). The **Fundamental Theorem of Calculus (FTC)** unites them, proving **accumulation and rate-of-change are inverse operations**.

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

**Part 1** shows differentiation and integration undo each other. **Part 2** is the practical powerhouse, eliminating tedious Riemann sums by using antiderivatives.

## Practical Power & Pitfalls

To compute $\int_0^2 x\, dx$ using **Part 2**, find the antiderivative $F(x) = \frac{x^2}{2}$ and evaluate at endpoints: $\left[\frac{x^2}{2}\right]_0^2 = 2 - 0 = 2$, matching a triangle's area exactly.

| Rule Aspect | Detail & Best Practice |
| --- | --- |
| **Endpoint Order** | Always upper minus lower: $F(b) - F(a)$ |
| **Sign Reversal** | Reversing limits flips the final sign |
| **Constant $+C$** | Cancels out during subtraction; any antiderivative works |

> **Common Pitfall:** Subtracting endpoints in the wrong order, or attempting to use Part 2 on indefinite integrals. The $+C$ constant always vanishes in the subtraction.

## Practice questions

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

### 1. Order the steps of evaluating a definite integral using the Fundamental Theorem.

**Answer:**

1. Find any antiderivative F of the integrand
2. Evaluate F at the upper limit b
3. Evaluate F at the lower limit a
4. Subtract to form F(b) - F(a)
5. Report the resulting number

**Why:** Antiderivative first, then the two endpoint values, then the subtraction in the order upper minus lower. Because any antiderivative works, the constant C cancels in the subtraction and never needs writing down.

Page: https://tryals.app/practice/computer-science-i/the-fundamental-theorem-of-calculus/order-the-steps-of-evaluating-a-definite-integral-using-the

### 2. The Fundamental Theorem connects the derivative and the integral, which look like unrelated operations. Why should differentiation and area-finding be inverse to each other?

A. Because both operations are defined as limits, and all limit operations are inverse to one another.
B. Because area and slope are numerically equal for most functions.
C. Because the area accumulated up to x grows at the height of the curve.
D. Because the integral is defined as the inverse of the derivative, making the theorem a definition.

**Answer:** C. Because the area accumulated up to x grows at the height of the curve.

**Why:** Push x forward by h and the extra area is a sliver of width h and height about f(x), so A′ = f, the area function is an antiderivative, and Part 2 follows since any other differs by a constant. Both objects are defined independently, one as a limit of Riemann sums and the other by differentiation; that they coincide is a theorem, and it is what made integrals computable.

Page: https://tryals.app/practice/computer-science-i/the-fundamental-theorem-of-calculus/the-fundamental-theorem-connects-the-derivative-and-the-integral

### 3. Euler's method has local error O(h²) per step and global error O(h). Where does the lost order go?

A. Into round-off error, which grows as the step shrinks
B. Into the initial condition, whose error propagates unchanged
C. Into the number of steps, about 1/h of them over a fixed interval
D. Nowhere, the two figures describe different quantities and are unrelated

**Answer:** C. Into the number of steps, about 1/h of them over a fixed interval

**Why:** h² per step times 1/h steps is h, the accumulation costs exactly one order. That is why Euler is first-order and halving the step only halves the error, and why RK4 evaluates the derivative four times per step to reach O(h⁴), a far better return on the extra work.

Page: https://tryals.app/practice/computer-science-i/the-fundamental-theorem-of-calculus/eulers-method-has-local-error-o-h-per-step-and-global-error-o-h

### 4. Complete the statement of the first part of the theorem.

**Answer:** If F(x) is defined as the integral of f from a fixed a up to x, then F'(x) equals **f(x)**.

**Why:** The derivative of the accumulated area is the height of the curve, f(x). This is the statement that differentiation undoes integration, and it is why velocity integrates to displacement and a rate integrates to a total.

Page: https://tryals.app/practice/computer-science-i/the-fundamental-theorem-of-calculus/complete-the-statement-of-the-first-part-of-the-theorem

### 5. Define F(x) as the definite integral of t^2 from t = 0 up to t = x. Using the first part of the theorem, compute F'(3).

**Answer:** 9

**Why:** F'(x) = x^2, so F'(3) = 9. Evaluating F(3) = 9 as well is a coincidence of this integrand; the point is that differentiating an accumulation gives the height back with no integration performed.

Page: https://tryals.app/practice/computer-science-i/the-fundamental-theorem-of-calculus/define-f-x-as-the-definite-integral-of-t-2-from-t-0-up-to-t-x

### 6. Select every statement that follows from the Fundamental Theorem of Calculus.

A. Accumulation and rate of change are inverse operations
B. The choice of antiderivative does not affect the answer
C. A definite integral can be computed without any summation
D. Every function has an antiderivative expressible in elementary terms
E. Reversing the limits of integration flips the sign of the answer

**Answer:** A. Accumulation and rate of change are inverse operations; B. The choice of antiderivative does not affect the answer; C. A definite integral can be computed without any summation; E. Reversing the limits of integration flips the sign of the answer

**Why:** The theorem makes evaluation a subtraction, makes the constant irrelevant, and forces F(a) - F(b) to be the negative of F(b) - F(a). It says nothing about whether a formula for the antiderivative exists, many do not, which is exactly why numerical methods survive.

Page: https://tryals.app/practice/computer-science-i/the-fundamental-theorem-of-calculus/select-every-statement-that-follows-from-the-fundamental-theorem-of