# Applications of Integration

Mathematics I · Integral Calculus · https://tryals.app/learn/mathematics-i/applications-of-integration

## The Reach of the Integral

Every application of integration shares one unifying idea: wherever a quantity **accumulates** over an extent, an integral computes the total by summing up infinitesimal pieces.

| Domain | What You Integrate | What It Yields |
| --- | --- | --- |
| **Kinematics** | **Velocity** $v(t)$ over time | **Displacement** (total distance) |
| **Economics** | **Marginal cost** over quantity | **Total cost** |
| **Physics** | **Force** $F(x)$ over distance | **Work** $W = \int F\, dx$ |

**From rates to totals.** The integral undoes the derivative. Integrating a **flow rate** yields total volume, just as integrating a **marginal cost** yields total cost. This is the Fundamental Theorem in action.

**Common pitfall:** Viewing these applications as separate formulas to memorize rather than one core idea. Recognizing the "sum up the little pieces" structure is what lets you set up any application correctly.

## Probability and Geometry

Continuous **probability** is fundamentally integral calculus. For a continuous random variable with **probability density** $f(x)$:

- The probability of landing in an interval is the **integral of the density** over that interval.
- The total integral over all possible values must equal **1**.
- The **mean** (expected value) is calculated as $\int x f(x)\, dx$.

**Geometry.** Area between curves, volumes of revolution, arc length, and surface area are all integrals of infinitesimal geometric slices.

The two operations of calculus form a complete cycle:

| Operation | What It Does |
| --- | --- |
| **Derivative** | Breaks a total into instantaneous rates |
| **Integral** | Reassembles a total from infinitesimal pieces |

## Practice questions

3 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. A differential equation like dN/dt = kN specifies a rate rather than a formula. Why does one such equation have infinitely many solutions, and what picks out the one you want?

A. Because the equation says nothing about where N starts.
B. Because each solution is valid on a different interval, and together they cover the domain.
C. Because the constant of integration is arbitrary and may be set to zero without loss.
D. Because differential equations are approximations, and the multiple solutions reflect the imprecision.

**Answer:** A. Because the equation says nothing about where N starts.

**Why:** The equation constrains the shape of the curve and says nothing about its scale, so a colony of 100 and one of 10,000 satisfy it with the same k. The constant is arbitrary in that the equation does not determine it, and it is not free to discard, since C = 0 gives the solution in which nothing exists. It carries the physical information the equation was never going to supply.

Page: https://tryals.app/practice/mathematics-i/applications-of-integration/a-differential-equation-like-dn-dt-kn-specifies-a-rate-rather-than

### 2. Select every quantity that is computed by accumulating infinitesimal pieces, that is, by an integral.

A. The slope of a tangent line
B. The work done by a variable force
C. The volume of a solid of revolution
D. The area between two curves
E. The instantaneous velocity at t = 3

**Answer:** B. The work done by a variable force; C. The volume of a solid of revolution; D. The area between two curves

**Why:** Work, area and volume are all totals assembled from infinitesimal pieces. Instantaneous velocity and the slope of a tangent are values at a single point, which is what derivatives compute, the two operations are inverse, so knowing which question is being asked tells you which tool applies.

Page: https://tryals.app/practice/mathematics-i/applications-of-integration/select-every-quantity-that-is-computed-by-accumulating-infinitesimal

### 3. dN/dt = kN says a population's growth is proportional to its size, and predicts unbounded exponential growth. Why does that model still earn its place given no population does that?

A. Because it applies to bacteria but not to larger organisms
B. Because it is accurate while resources are unlimited
C. Because populations do in fact grow exponentially without limit
D. Because it is the only differential equation that can be solved exactly

**Answer:** B. Because it is accurate while resources are unlimited

**Why:** It is right for the early phase and wrong later, and the shape of its failure names the missing ingredient, a carrying capacity, which the logistic equation adds. A simple model that fails informatively is often more useful than a complicated one that fits, because you can see what it left out.

Page: https://tryals.app/practice/mathematics-i/applications-of-integration/dn-dt-kn-says-a-populations-growth-is-proportional-to-its-size