# Integrals as accumulation

Physics I · Calculus of a Single Variable · https://tryals.app/learn/physics-i/integrals-as-accumulation

## Definite Integrals as Accumulation

The **definite integral** measures signed area under a curve, built by adding infinite slices via **Riemann sums**: 

$$\int_{a}^{b} f(x)\,dx \;=\; \lim_{n\to\infty}\sum_{i=1}^{n} f(x_{i}^{*})\,\Delta x$$

Here, $[a,b]$ is partitioned into $n$ sub-intervals of width $\Delta x = \frac{b-a}{n}$, with a sample point $x_i^*$ in each.

| Property | Formula |
| --- | --- |
| **Linearity** | $\int_a^b [\alpha f + \beta g]\,dx = \alpha\int f + \beta\int g$ |
| **Additivity** | $\int_a^c f\,dx = \int_a^b f\,dx + \int_b^c f\,dx$ |
| **Average Value** | $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$ |

**Common pitfall:** A definite integral is a signed number, not pure area. Regions below the axis count negative; total area needs splits at zero crossings.

## Core Rules & Applications

Integrals accumulate rates of change into total amounts. In physics, the **Work** done by a variable force is $W=\int_{a}^{b}F(x)\,dx$.

To compute these, master basic **antiderivatives**:

| Function | Antiderivative | Edge Case |
| --- | --- | --- |
| **Power rule** $x^n$ | $\frac{x^{n+1}}{n+1} + C$ | $n \neq -1$ |
| **Reciprocal** $1/x$ | $\ln|x| + C$ | Domain excludes $0$ |
| **Exponential** $e^x$ | $e^x + C$ | Always true |

The constant $C$ accounts for the family of all valid antiderivatives in indefinite integration.

## Practice questions

6 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. $\int_a^a f(x) dx$ = 0 for any continuous function f.

**Answer:** True

**Why:** A definite integral over a degenerate interval [a, a] has zero width and hence zero area.

Page: https://tryals.app/practice/physics-i/integrals-as-accumulation/a-a-f-x-dx-0-for-any-continuous-function-f

### 2. A definite integral of velocity gives displacement, not distance travelled. Why do the two differ when the object reverses direction?

A. Because displacement is measured in different units from distance
B. Because the integral adds signed contributions, so reversals subtract
C. Because reversal makes the velocity function discontinuous
D. Because the integral is only an approximation to the true distance

**Answer:** B. Because the integral adds signed contributions, so reversals subtract

**Why:** The sign is information, not a defect. Signed accumulation is what makes the integral report net change; distance is a different question requiring $|v|$.

Page: https://tryals.app/practice/physics-i/integrals-as-accumulation/a-definite-integral-of-velocity-gives-displacement-not-distance

### 3. Which are valid properties of the definite integral?

A. $\int_a^b f dx$ = −$\int_b^a f dx$
B. $\int_a^b (f\cdot g) dx$ = ($\int_a^b f dx$)·($\int_a^b g dx$)
C. $\int_a^b (f + g) dx$ = $\int_a^b f dx$ + $\int_a^b g dx$
D. $\int_a^b c\cdot f dx$ = c·$\int_a^b f dx$

**Answer:** A. $\int_a^b f dx$ = −$\int_b^a f dx$; C. $\int_a^b (f + g) dx$ = $\int_a^b f dx$ + $\int_a^b g dx$; D. $\int_a^b c\cdot f dx$ = c·$\int_a^b f dx$

**Why:** the second option is false: the integral of a product is not the product of integrals.

Page: https://tryals.app/practice/physics-i/integrals-as-accumulation/which-are-valid-properties-of-the-definite-integral

### 4. Order the steps in constructing a Riemann sum to define the definite integral.

**Answer:**

1. Divide [a, b] into n equal subintervals of width $\Delta x = (b - a)/n$
2. Choose a sample point xᵢ* in each subinterval
3. Form the Riemann sum: Σᵢ f(xᵢ*)·$\Delta x$
4. Take the limit as $n \to \infty$ to obtain $\int_a^b f(x) dx$

**Why:** The definite integral is the limit of finer and finer Riemann approximations.

Page: https://tryals.app/practice/physics-i/integrals-as-accumulation/order-the-steps-in-constructing-a-riemann-sum-to-define-the-definite

### 5. Your car’s **odometer** ticks upward as you drive. Which calculus object is the odometer physically computing from your speed?

A. The derivative of speed over time
B. The maximum speed sustained so far
C. The integral of speed over time
D. The average of the speedometer readings

**Answer:** C. The integral of speed over time

**Why:** The odometer accumulates $v\,dt$ moment by moment: a running Riemann sum, i.e. $\int v\,dt$. Speedometer (derivative) and odometer (integral) driving in tandem are the fundamental theorem bolted to your dashboard.

Page: https://tryals.app/practice/physics-i/integrals-as-accumulation/your-cars-odometer-ticks-upward-as-you-drive-which-calculus-object

### 6. Water flows **into** a tank at rate $r(t) > 0$, then **out** at rate $|r(t)|$ when $r(t) < 0$. What does $\int_0^T r(t)\,dt$ report?

A. The net change in water volume
B. The average flow rate
C. The final volume in the tank
D. The total volume that passed through, ignoring direction

**Answer:** A. The net change in water volume

**Why:** Signed area = net change: outflow intervals subtract. To get the final amount you must add the initial state, $V(T) = V(0) + \int_0^T r\,dt$, the accumulation form of the fundamental theorem.

Page: https://tryals.app/practice/physics-i/integrals-as-accumulation/water-flows-into-a-tank-at-rate-r-t-0-then-out-at-rate-r-t