Unit 10 · Term 2

Applications of the Integral

Once the integral is "add up infinitely many tiny pieces," geometric and physical quantities follow: area, volume, average value, and arc length. The skill is setting up the right integral.

By the end you can

Find the area between two curves.
Compute volumes of revolution by disks/washers and shells.
Find the average value of a function.
Set up an arc-length integral.

Why it matters & where it comes from

Why it exists

Once you can integrate, geometry and physics open up — areas, volumes, lengths, and averages all become single integrals.

In the real world

Volumes of revolution show up in manufacturing and 3D printing; CT and MRI scanners reconstruct the body by integrating along rays; arc length is path planning and GPS. A signal's average value gives its DC component, while its RMS value is the square root of the average of its square.

Story & open questions

Archimedes found the volume of a sphere this way and was so proud he asked for a sphere-in-a-cylinder on his tombstone. In 1615 Kepler worked out the volumes of wine barrels — an early, very practical burst of integration.

Setup comes before calculus

Applications of the integral are won or lost before the antiderivative. First identify the tiny piece being added: a vertical strip, a washer, a shell, a cross-section, or a small arc-length segment. Then translate that piece into an integral.

Setup routine

Draw the region and find intersections.

Decide the slice direction: vertical slices use $dx$, horizontal slices use $dy$.

Write the size of one slice: height, radius, area, or length element.

Integrate over the interval covered by the slices.

Common formulas as slice stories

Area. A thin rectangle has area $(\text{top}-\text{bottom})\,dx$.
Disk/washer. A thin circular slice has area $\pi(R^2-r^2)$ and thickness $dx$ or $dy$.
Shell. A thin shell has circumference $2\pi r$, height $h$, and thickness $dx$ or $dy$.
Arc length. A tiny segment has length $\sqrt{1+(f'(x))^2}\,dx$.

Area between curves

Formula

If $f\ge g$ on $[a,b]$, the area between them is $\displaystyle\int_a^b\big(f(x)-g(x)\big)\,dx.$

Example

Between $y=x^2$ and $y=2-x^2$: they meet at $x=\pm1$, and $\int_{-1}^{1}(2-2x^2)\,dx=\big[2x-\tfrac{2}{3}x^3\big]_{-1}^{1}=\tfrac83.$

Volumes of revolution

Two methods

Disks/washers (slices $\perp$ to the axis): $V=\displaystyle\int_a^b\pi\big(R^2-r^2\big)\,dx.$ Shells (slices $\parallel$): for a region in $x\ge0$ rotated about the $y$-axis, $V=\displaystyle\int_a^b 2\pi x\,h(x)\,dx$. In general, use each slice's distance from the axis as its radius.

Example

Revolve $y=\sqrt{x}$, $0\le x\le4$, about the $x$-axis: $V=\int_0^4\pi(\sqrt{x})^2\,dx=\pi\int_0^4 x\,dx=8\pi.$

Average value

Definition

The average value of $f$ on $[a,b]$ is $\bar f=\dfrac{1}{b-a}\displaystyle\int_a^b f.$ The Mean Value Theorem for integrals says a continuous $f$ attains this value somewhere.

Arc length

Formula

If $f'$ is continuous on $[a,b]$, the length of $y=f(x)$ over $[a,b]$ is $L=\displaystyle\int_a^b\sqrt{1+\big(f'(x)\big)^2}\,dx.$

Common pitfalls

• Find intersection points before integrating; check which curve is on top.
• For washers, subtract the inner radius squared, not the radii then squared.
• Match the slicing method to the axis to avoid impossible integrals.

Practice

Problems adapted from the MAT137 question bank by Alfonso Gracia-Saz, used under CC BY-SA 4.0. Write full solutions before opening the answer.

Practice 1 · Area

Find the area between $y=x$ and $y=x^2$.

Answer

They meet at $0$ and $1$; $\int_0^1(x-x^2)\,dx=\tfrac12-\tfrac13=\tfrac16.$

Practice 2 · Shells

Revolve $y=x^2$, $0\le x\le1$, about the $y$-axis using shells.

Answer

$V=\int_0^1 2\pi x\cdot x^2\,dx=2\pi\cdot\tfrac14=\tfrac{\pi}{2}.$

Practice 3 · Average value

Average value of $f(x)=\sin x$ on $[0,\pi]$.

Answer

$\dfrac{1}{\pi}\int_0^\pi \sin x\,dx=\dfrac{2}{\pi}.$

Practice 4 · Disk formula

Let $f$ be positive on $[a,b]$. Write the volume when the region under $f$ is rotated about the $x$-axis.

Answer

$V=\displaystyle\int_a^b \pi\,[f(x)]^2\,dx$ — a stack of disks of radius $f(x)$.

Practice 5 · Sphere

Rotate the circle $x^2+y^2=R^2$ about the $x$-axis and find the sphere's volume by disks.

Solution

$V=\int_{-R}^{R} \pi(R^2-x^2)\,dx=\pi\!\left[R^2x-\tfrac{x^3}{3}\right]_{-R}^{R} =\tfrac43\pi R^3.$

Practice 6 · Pyramid

Find the volume of a pyramid of height $H$ with a square base of side $L$, using slices parallel to the base.

Solution

A slice a distance $y$ from the apex has side $Ly/H$, so $V=\int_0^H\!\left(\tfrac{Ly}{H}\right)^2dy =\tfrac{L^2H}{3}.$

Practice 7 · Washer

Rotate the region between $y=\sqrt{x}$ and $y=x$ about the $x$-axis.

Answer

$V=\int_0^1\pi\!\left[x-x^2 \right]dx=\pi\!\left(\tfrac12-\tfrac13\right)=\tfrac{\pi}{6}.$

Practice 8 · Washer

Rotate the first-quadrant region between $y=x^3$ and $y=\sqrt{32x}$ about the $x$-axis.

Solution

They meet at $x=2$ ($x^5=32$). $V=\int_0^2\pi\!\left[32x-x^6\right]dx =\pi\!\left[16x^2-\tfrac{x^7}{7}\right]_0^2=\tfrac{320\pi}{7}.$

Expanded topic checklist

Course coverage. This unit is part of the complete MAT137 topic map. It should cover the following items before you move on:

Area and signed area. Set up integrals for area between curves using top-minus-bottom or right-minus-left.
Volumes. Use disks, washers, and shells with a clear axis of rotation and radius function.
Average value. Interpret $f_{avg}=rac1{b-a}\int_a^b f$ and connect it to accumulation.
Physical applications. Translate density, work, and rate problems into integrals with units.

Textbook/source alignment

Use the Alfonso Gracia-Saz MAT137 materials as the primary local spine, with the current UofT single-variable calculus topic split as the completeness check. The added items are original study prompts, not copied textbook exercises.

Core theorem map

The application spine is: slice the object, write the small piece, integrate the small pieces over the correct interval, and check units.

Study routine. Before computing, name the variable of integration and the geometry of one slice; if that sentence is vague, the integral is probably wrong.

Deep practice extensions

Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.

Extension 1 · Area between curves

Find the area between $y=x$ and $y=x^2$ on $[0,1]$.

Guide

Top minus bottom is $x-x^2$, so the area is $\int_0^1(x-x^2)dx=1/2-1/3=1/6$.

Extension 2 · Washer setup

Set up the volume from rotating the region under $y=\sqrt{x}$ on $[0,4]$ about the $x$-axis.

Guide

Washers have radius $\sqrt{x}$, so $V=\pi\int_0^4(\sqrt{x})^2dx=\pi\int_0^4 x dx$.

Extension 3 · Average value

Find the average value of $f(x)=x^2$ on $[0,3]$.

Guide

Compute $(1/3)\int_0^3 x^2dx=(1/3)(9)=3$.

Extension 4 · Units

If density is kg/m and position is measured in meters, what units should $\int_a^b ho(x)dx$ have?

Guide

The integrand contributes kg/m and $dx$ contributes m, so the integral has units kg: total mass.

← Unit 09Integration methods Unit 11 →Sequences