Table of Contents
Using Integrals to Measure and Accumulate
Indefinite and definite integrals give us a way to describe how quantities accumulate. In this chapter we focus on what you can do with integrals: how they measure area, volume, total change, and many other “totals” built from rates or densities.
You should already be familiar with the idea of a definite integral as a limit of Riemann sums and with basic computation of integrals. Here we concentrate on interpreting integrals in real-world and geometric situations.
General Interpretation: Integral as Accumulated Quantity
Many applications share a common pattern:
- You have a quantity that is spread out over an interval (in space or time).
- You know its density or rate at each point.
- You want the total amount over the whole interval.
If $x$ is a variable (time, length, etc.) and $f(x)$ is a rate or density (per unit of $x$), then the total accumulated quantity from $x = a$ to $x = b$ is
$$
\int_a^b f(x)\,dx.
$$
You can think of $f(x)\,dx$ as the contribution from a tiny piece of width $dx$. The integral adds up all those tiny contributions.
Common language pairs:
- rate $\to$ total change,
- density $\to$ total mass/charge/people/etc.,
- cross-sectional area $\to$ volume.
We now examine two of the most important geometric applications: area in the plane and volume of solids.
Area Between Curves
You already know that if $f(x) \ge 0$ on $[a,b]$, then the definite integral
$$
\int_a^b f(x)\,dx
$$
represents the (signed) area under the graph of $f$ and above the $x$–axis. We now extend this to the area between two curves.
Area Between a Top and Bottom Function
Suppose on the interval $[a,b]$:
- $f(x)$ is the top function,
- $g(x)$ is the bottom function,
so that $f(x) \ge g(x)$ for all $x$ in $[a,b]$.
Then the area of the region bounded vertically by $y=f(x)$ and $y=g(x)$, and between $x=a$ and $x=b$, is
$$
A = \int_a^b \bigl[f(x) - g(x)\bigr]\,dx.
$$
Reasoning: For each small strip of width $dx$, the height is “top minus bottom” $f(x)-g(x)$, so area of the thin strip is $(f(x)-g(x))\,dx$. The integral adds all these strip areas from $a$ to $b$.
Finding the Limits of Integration
In many problems, the intersection points of the curves become the limits of integration. You often:
- Solve $f(x) = g(x)$ to find intersection $x$–values.
- Check which function is on top between those intersections.
- Integrate top minus bottom.
If the order of top/bottom changes on the interval, you must:
- break the integral at the $x$ where they switch order,
- or reverse the roles of $f$ and $g$ where necessary,
to ensure each part is “top minus bottom.”
Area with Respect to $y$ (Horizontal Strips)
Sometimes curves are more easily described as $x$ as a function of $y$ (i.e., $x = f(y)$). Then the region may be bounded by a right curve and a left curve.
If:
- $x = R(y)$ is the rightmost boundary,
- $x = L(y)$ is the leftmost boundary,
and $R(y) \ge L(y)$ on $[c,d]$,
then the area is
$$
A = \int_c^d \bigl[ R(y) - L(y) \bigr]\,dy.
$$
Here, each horizontal strip has height $dy$ and length $R(y)-L(y)$.
Working with $y$ instead of $x$ is especially useful when:
- the region is naturally “stacked” horizontally,
- or expressing $y$ as a function of $x$ is hard or impossible.
Volumes of Solids of Revolution
Integrals also allow us to compute volumes of many three-dimensional shapes, especially those obtained by revolving a region around an axis. The general strategy is to slice the solid into many thin pieces, find the volume of a typical slice, and integrate.
There are three main methods you will encounter:
- disk method,
- washer method,
- shell method.
Each method corresponds to a different way of slicing the solid.
Disk Method (No Hollow Core)
Use the disk method when revolving a region that touches the axis of rotation, so the cross-sections perpendicular to the axis are solid disks.
Revolving Around the $x$–Axis
Suppose $y = f(x) \ge 0$ on $[a,b]$, and the region between the curve and the $x$–axis is revolved around the $x$–axis. The resulting solid has cross-sections perpendicular to the $x$–axis that are circles (disks).
At position $x$:
- radius $= f(x)$,
- area of disk $= \pi [f(x)]^2$.
The volume is
$$
V = \int_a^b \pi [f(x)]^2\,dx.
$$
Revolving Around the $y$–Axis
If instead you have $x = g(y) \ge 0$ for $y$ in $[c,d]$ and revolve the region between this curve and the $y$–axis around the $y$–axis, then the cross-sectional disks are perpendicular to the $y$–axis:
At height $y$:
- radius $= g(y)$,
- area $= \pi [g(y)]^2$,
so
$$
V = \int_c^d \pi [g(y)]^2\,dy.
$$
Washer Method (Hollow Core)
Use the washer method when the region being revolved does not touch the axis of rotation, creating a hole in the middle, like a washer (a disk with a circular hole).
The cross-section perpendicular to the axis is now an annulus (ring). Each washer is described by:
- an outer radius $R$,
- an inner radius $r$.
Area of a washer:
$$
\text{Area} = \pi R^2 - \pi r^2 = \pi\bigl(R^2 - r^2\bigr).
$$
Revolving Around the $x$–Axis
Suppose the region between $y = f(x)$ and $y = g(x)$ on $[a,b]$ is revolved around the $x$–axis, with $f(x)$ farther from the $x$–axis than $g(x)$ (so $f(x) \ge g(x) \ge 0$):
- Outer radius: $R(x) = f(x)$,
- Inner radius: $r(x) = g(x)$.
Then the volume is
$$
V = \int_a^b \pi\bigl(R(x)^2 - r(x)^2\bigr)\,dx
= \int_a^b \pi\bigl(f(x)^2 - g(x)^2\bigr)\,dx.
$$
Similarly, for revolution around the $y$–axis, you express radii in terms of $y$ and integrate with respect to $y$.
Shell Method (Cylindrical Shells)
With the disk or washer method, slices are perpendicular to the axis of rotation. The shell method uses slices that are parallel to the axis, forming cylindrical shells when revolved.
A typical shell:
- has radius $r$,
- height $h$,
- and thickness $dx$ or $dy$.
If you unroll a thin cylindrical shell, its lateral surface area is approximately
$$
\text{Lateral area} \approx 2\pi r \cdot h.
$$
Multiplying by the shell thickness gives the shell volume:
$$
dV \approx 2\pi r h \, (\text{thickness}).
$$
Shell Method Around the $y$–Axis (Integrate with Respect to $x$)
Consider a region between $x = a$ and $x = b$, bounded above and below by $y$–values, and revolve it around the $y$–axis.
Take a vertical strip at position $x$ (width $dx$). When revolved about the $y$–axis, this strip generates a cylindrical shell.
- Shell radius: $r(x) = x$ (distance from $x$ to the $y$–axis),
- Shell height: determined by the vertical extent of the strip, say $h(x)$,
- Shell thickness: $dx$.
Then the volume is
$$
V = \int_a^b 2\pi r(x) h(x)\,dx.
$$
In many problems, $h(x)$ is “top function minus bottom function” in $y$:
$$
h(x) = f(x) - g(x),
$$
so
$$
V = \int_a^b 2\pi x\,[f(x) - g(x)]\,dx.
$$
Shell Method Around the $x$–Axis (Integrate with Respect to $y$)
If you revolve a region around the $x$–axis and you prefer to integrate with respect to $y$, you use horizontal strips of thickness $dy$.
At height $y$:
- radius: $r(y) = y$ (distance to the $x$–axis),
- height: the horizontal length of the region at that $y$, say $h(y)$.
Then
$$
V = \int_c^d 2\pi r(y) h(y)\,dy = \int_c^d 2\pi y\,h(y)\,dy.
$$
Choosing Between Disk/Washer and Shell
Often both methods are theoretically possible; one will be easier.
Typical preferences:
- If the solid is revolved around a vertical axis like the $y$–axis:
- vertical slices (shell method, integrate in $x$),
- or horizontal slices (disk/washer, integrate in $y$).
- If revolved around a horizontal axis like the $x$–axis:
- horizontal slices (shell method, integrate in $y$),
- or vertical slices (disk/washer, integrate in $x$).
Choose the direction of slicing that:
- makes the radii and heights easy to express,
- avoids solving difficult equations for inverse functions,
- and minimizes how many times you must split the integral.
Physical and Real-World Accumulations
The same logic behind area and volume extends to many physical contexts. The pattern is always:
Total quantity $= \int$ (density or rate) $\times$ (small piece).
Here are some common types.
Total Change from a Rate
If $r(t)$ is the rate of change of some quantity $Q$ with respect to time (for example, velocity, flow rate, production rate), then the net change in $Q$ from $t=a$ to $t=b$ is
$$
\Delta Q = \int_a^b r(t)\,dt.
$$
If $Q(a)$ is known, you get
$$
Q(b) = Q(a) + \int_a^b r(t)\,dt.
$$
Examples include:
- position from velocity,
- amount of water from flow rate,
- total cost from marginal cost.
Accumulating Density over a Line
Suppose a thin rod lies along the $x$–axis from $x=a$ to $x=b$, and its linear density (mass per unit length) is $\lambda(x)$ (for example, in kg/m). A tiny piece of length $dx$ at position $x$ has mass approximately $\lambda(x)\,dx$.
The total mass is
$$
M = \int_a^b \lambda(x)\,dx.
$$
Analogously:
- charge from linear charge density,
- population from people per kilometer along a road,
- fuel along a pipeline.
Density over an Area or Volume
In higher dimensions:
- If $\rho(x,y)$ is a surface density (mass per unit area), then the mass over a region $R$ is
$$
M = \iint_R \rho(x,y)\,dA.
$$ - If $\rho(x,y,z)$ is a volume density (mass per unit volume), then the mass over a solid region $D$ is
$$
M = \iiint_D \rho(x,y,z)\,dV.
$$
These are examples of multiple integrals, which you will study more fully in multivariable calculus, but the underlying idea is the same: integrate density over the region.
Work Done by a Variable Force (Preview)
If a force $F(x)$ acts along a line and can vary with position, then the work done in moving an object from $x=a$ to $x=b$ is
$$
W = \int_a^b F(x)\,dx.
$$
This fits the pattern “work $=$ force $\times$ distance,” with force varying over distance.
In other contexts, a similar integral describes:
- pumping liquids (integrate weight over height),
- stretching springs (force depending on extension).
These topics are typically developed further in more advanced courses or chapters, but the integral concept is the same.
Summary
- A definite integral represents accumulated quantity: rate or density times a small piece, summed over an interval.
- The area between curves is found by integrating “top minus bottom” (with respect to $x$) or “right minus left” (with respect to $y$).
- Volumes of solids of revolution are computed by:
- disk method when there is no hole,
- washer method when there is a hole,
- shell method using cylindrical shells parallel to the axis of rotation.
- Many physical applications—total change, mass from density, work from force—are all examples of “integral as accumulated quantity.”
Later topics will extend these ideas to more dimensions and more complex settings, but the fundamental interpretation of the integral as an accumulator remains the same.