# Integrated single-variable calculus problems

Physics I · Calculus of a Single Variable · https://tryals.app/learn/physics-i/integrated-single-variable-calculus-problems

## 1. Synthesis & Setup

Calculus problems combine limits, derivatives, integrals, and series. **Model the system** by translating physical descriptions into $f(x)$ with a clear domain.

| Step | Action | Focus |
| --- | --- | --- |
| 1. Model | Define $f(x)$ | Domain & constraints |
| 2. Differentiate | Find slopes | Critical points |
| 3. Integrate | Accumulate | Area, work, totals |
| 4. Approximate | Use Taylor series | Complex functions |

**Common pitfall:** The hardest part is the setup—naming variables and writing constraints. If the setup is wrong, perfect calculus cannot save it.

**Strategy:** Always sketch first. A good graph reveals domain restrictions, symmetry, and extrema before you compute a single derivative.

## 2. Core Formulas & Toolkit

Apply these standard formulas for geometric and complex quantities.

**Area between curves:**
$$A = \int_{a}^{b}\bigl|f(x)-g(x)\bigr|\,dx$$

**Arc length:**
$$L = \int_{a}^{b}\sqrt{1+\bigl(f'(x)\bigr)^{2}}\,dx$$

**Complex-number toolkit:**
- **Modulus:** $|z| = \sqrt{a^{2}+b^{2}}$
- **Conjugate:** $\overline{z} = a - bi$, meaning $z\,\overline{z}=|z|^{2}$
- **Roots of unity:** The $n$-th roots of $1$ are $e^{2\pi i k/n}$ for $k=0,\ldots,n-1$.

**Validate** your final answer by checking units, signs, boundary behavior, and limiting cases.

## Practice questions

7 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. Which statements are true for $f(x) = \sin(x)$?

A. f is an odd function
B. $\int_{0}^{2\pi} \sin(x) dx$ = 1
C. f achieves its maximum value of 1 at $x = \pi/2$
D. $f'(x) = \cos(x)$

**Answer:** A. f is an odd function; C. f achieves its maximum value of 1 at $x = \pi/2$; D. $f'(x) = \cos(x)$

**Why:** $\int_{0}^{2\pi} \sin x dx$ = 0, not 1 (symmetric positive and negative areas cancel).

Page: https://tryals.app/practice/physics-i/integrated-single-variable-calculus-problems/which-statements-are-true-for-f-x-sin-x

### 2. One curve, four instruments. For $s(t)$ = position of a particle, match each question to the tool that answers it.

**Answer:**

- How fast at instant $t$? → Derivative $s'(t)$
- When does it momentarily stop? → Solve $s'(t) = 0$
- Net displacement over $[a,b]$? → $\int_a^b s'(t)\,dt$
- Is it speeding up or slowing? → Compare signs of $s'$ and $s''$

**Why:** A single motion problem exercises the whole toolkit: derivatives for instants, integrals for totals, sign analysis for behaviour. Fluency is knowing **which tool the question is secretly asking for**: the skill this unit was building.

Page: https://tryals.app/practice/physics-i/integrated-single-variable-calculus-problems/one-curve-four-instruments-for-s-t-position-of-a-particle-match

### 3. Optimisation, related rates and accumulation problems all reduce to a small number of calculus moves. How can you tell which move a physical problem needs?

A. The number of variables appearing in the problem statement
B. What the problem asks for relative to what it gives
C. Whether the problem involves motion or a static configuration
D. Whether the units are SI or otherwise

**Answer:** B. What the problem asks for relative to what it gives

**Why:** Matching the given to the wanted is the whole of problem selection. Units make it nearly mechanical: given m/s and wanting m, you integrate.

Page: https://tryals.app/practice/physics-i/integrated-single-variable-calculus-problems/optimisation-related-rates-and-accumulation-problems-all-reduce-to-a

### 4. The same limit can be attacked by factoring, by conjugates, by L'Hopital's rule or by a series expansion. Why do textbooks teach all four instead of the most general one?

A. Because historical convention requires all four to be covered
B. Because the general methods have preconditions the others lack
C. Because the four methods give slightly different answers that are averaged
D. Because L'Hopital's rule is not rigorous and needs confirmation

**Answer:** B. Because the general methods have preconditions the others lack

**Why:** Generality is bought with preconditions. Applying L'Hopital to a determinate form gives a wrong answer confidently, which is why recognising the form comes before choosing the tool.

Page: https://tryals.app/practice/physics-i/integrated-single-variable-calculus-problems/the-same-limit-can-be-attacked-by-factoring-by-conjugates-by

### 5. Complete the statement with the correct term.

**Answer:** The inflection point of $f(x) = x^{3} - 3x$ is found by solving f''(x) = **0**.

**Why:** $f''(x) = 6x$. Setting $6x = 0$ gives $x = 0$. Concavity changes sign there: $f'' < 0$ for $x < 0$, $f'' > 0$ for $x > 0$.

Page: https://tryals.app/practice/physics-i/integrated-single-variable-calculus-problems/complete-the-statement-with-the-correct-term

### 6. Find the area between $y = x$ and $y =  x^{2}$ on [0, 1]. **Set the slider to your answer.**

**Answer:** 0.1667 (within ±0.005)

**Why:** $\int_{0}^{1} (x - x^{2}) dx$ = [$x^{2}/2$ − $x^{3}/3$]₀¹ = 1/2 − 1/3 = 1/6 $\approx 0.1667$.

Page: https://tryals.app/practice/physics-i/integrated-single-variable-calculus-problems/find-the-area-between-y-x-and-y-x-on-0-1-set-the-slider-to

### 7. Differentiation and integration are inverse operations, yet an indefinite integral carries a $+C$ and a derivative does not. What does that asymmetry record?

A. That the constant is a convention which may be omitted safely
B. That integration is less precise than differentiation
C. That differentiation destroys information about constants
D. That integration requires limits while differentiation does not

**Answer:** C. That differentiation destroys information about constants

**Why:** A non-injective operation cannot have a clean inverse. The $+C$ is exactly the information destroyed, and an initial condition is what supplies it again.

Page: https://tryals.app/practice/physics-i/integrated-single-variable-calculus-problems/differentiation-and-integration-are-inverse-operations-yet-an