Applications of Integrals

1. Introduction

The definite integral began as a way to measure area under a curve, but its real power is far broader: it is a machine for accumulating infinitely many infinitesimal contributions into a single total. Whenever a quantity can be sliced into tiny pieces that are each approximately a simple shape — a thin rectangle, a disk, a cylindrical shell — the integral sums those pieces exactly in the limit. This "slice, approximate, sum, take the limit" idea is the heart of applied integration.

In this article we apply integration to geometry and beyond: the area between curves, the volume of solids of revolution by disks, washers, and shells, volumes by general cross-section, arc length, the average value of a function, and the Mean Value Theorem for Integrals. Each application is a variation on the same template, and learning to recognize the right slicing direction is the key skill.

The Fundamental Theorem of Calculus does the heavy lifting at the end: once we have set up the correct integral, evaluating it is just antidifferentiation. The challenge — and the focus here — is the setup.

Every formula in this chapter — area, volume, arc length, average value — arises from the same Riemann sum template: $\sum (\text{slice quantity})\cdot\Delta x \to \int (\text{slice quantity})\,dx$. Learning to draw the representative slice and label its dimensions correctly is more important than memorizing individual formulas. The Mean Value Theorem for Integrals guarantees that the average value of a continuous function is actually attained at some point in the interval.

2. Core Concepts

2.1 The Integral as Accumulation

The definite integral $\int_a^b f(x)\,dx$ is the limit of Riemann sums $\sum f(x_i^*)\,\Delta x$. Each term is the area of a thin rectangle; the limit accumulates net signed area (regions below the axis count negatively). Every application follows: write the contribution of one infinitesimal slice, then integrate.

2.2 Area Between Two Curves

If $f(x) \ge g(x)$ on $[a,b]$, a vertical strip at $x$ has height $f(x) - g(x)$ and width $dx$: $A = \int_a^b \big[f(x) - g(x)\big]\,dx.$ Always (top) minus (bottom). Split at intersection points when curves cross. Horizontal strips use (right) minus (left) with respect to $y$.

2.3 Volumes by Cross-Section

If cross-sectional area perpendicular to the $x$-axis is $A(x)$: $V = \int_a^b A(x)\,dx.$ This generalizes disks, washers, and solids with square or semicircular cross-sections.

2.4 Disk and Washer Methods

Rotating about an axis, slices perpendicular to the axis give circular cross-sections.

Disk (region touches axis): $V = \pi\int_a^b [R(x)]^2\,dx$.

Washer (gap between region and axis): $V = \pi\int_a^b \big([R(x)]^2 - [r(x)]^2\big)\,dx$.

2.5 Cylindrical Shell Method

Slices parallel to the axis of rotation produce shells. A shell at $x$ has radius (distance to axis) and height $h(x)$: $V = 2\pi\int_a^b (\text{radius})(\text{height})\,dx \quad \text{(rotation about $y$-axis)}.$ Often easier when rotating a $y = f(x)$ region about the $y$-axis.

2.6 Arc Length

For smooth $y = f(x)$ on $[a,b]$, summing infinitesimal hypotenuses gives $L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx.$ Parametric curves $\mathbf{r}(t) = \langle x(t), y(t)\rangle$ use $L = \int \sqrt{[x'(t)]^2 + [y'(t)]^2}\,dt$.

2.7 Average Value and MVT for Integrals

The average value of $f$ on $[a,b]$ is $\bar f = \frac{1}{b-a}\int_a^b f(x)\,dx.$ The Mean Value Theorem for Integrals: if $f$ is continuous on $[a,b]$, then some $c \in [a,b]$ satisfies $f(c) = \bar f$.

Proof sketch: By the EVT, $f$ attains min $m$ and max $M$ on $[a,b]$. Then $m(b-a) \le \int_a^b f \le M(b-a)$, so $m \le \bar f \le M$. By the IVT, $f(c) = \bar f$ for some $c$.

2.8 Work and Fluid Force (Preview)

Work done by a variable force $W = \int_a^b F(x)\,dx$. Hydrostatic force on a submerged plate uses pressure $p = \rho g h$ integrated over depth.

2.9 Pappus's Centroid Theorems (Preview)

The volume of a solid of revolution equals $2\pi\bar d \cdot A$, where $\bar d$ is the distance from the centroid of the region to the axis and $A$ is the area of the region. This can shortcut certain volume problems.

2.10 Center of Mass (Centroid)

For a lamina with density $\rho(x)$ on $[a,b]$, the $x$-coordinate of the centroid is $\bar x = \frac{\int_a^b x\,\rho(x)\,dx}{\int_a^b \rho(x)\,dx}.$ For uniform density, $\rho$ cancels and $\bar x = \frac{1}{A}\int x\,dA$, where $A$ is the area.

2.11 Net Change from Rate

If $F'(x) = f(x)$, then the net change of $F$ on $[a,b]$ is $\int_a^b f(x)\,dx = F(b) - F(a)$. This interprets the integral of a rate (velocity, flow rate, marginal cost) as total accumulated change — a direct FTC application.

2.12 Comparison of Disk, Washer, and Shell Methods

Disks/washers slice perpendicular to the axis; shells slice parallel. Shells avoid solving for $x$ in terms of $y$ when rotating about the $y$-axis. Washers handle regions with a hole relative to the axis. The correct choice minimizes algebra.