Chapter 6 โ Applications of Integration
Slice, approximate, add up, take the limit โ one idea that turns the definite integral into a tool for areas, volumes, lengths, work and balance points.
In Calculus I the definite integral was born as the limit of a sum of thin rectangles under a curve. It would be easy to leave it there, as "the area under a graph". But the rectangles were never really about area. They were about a much more general recipe: when a quantity is hard to measure all at once, break it into many small pieces that are each easy to estimate, add the estimates, and let the pieces shrink. Whatever the quantity โ the volume of a fuel tank, the length of a cable hanging between two pylons, the energy needed to pump out a reservoir โ if it can be cut into pieces like this, it is an integral.
This chapter applies that recipe again and again. Engineers use it to size the concrete in a tapered column and the steel in a curved beam; physicians use it to estimate a tumour's volume from a stack of scan images; economists use it to measure the gap between what consumers are willing to pay and what they actually pay. The calculus is always the same four steps โ only the shape of the small piece changes.
| Quantity | Small piece | Size of one piece | Integral |
|---|---|---|---|
| Area between curves (6.1) | thin rectangle | $\left[f(x)-g(x)\right]\Delta x$ | $\int_a^b\left[f(x)-g(x)\right]dx$ |
| Volume (6.2) | disk or washer | $\pi\left(R^2-r^2\right)\Delta x$ | $\int_a^b\pi\left(R^2-r^2\right)dx$ |
| Volume (6.3) | cylindrical shell | $2\pi x\,f(x)\,\Delta x$ | $\int_a^b2\pi x\,f(x)\,dx$ |
| Arc length (6.4) | short line segment | $\sqrt{1+\left[f^\prime(x)\right]^2}\,\Delta x$ | $\int_a^b\sqrt{1+\left[f^\prime(x)\right]^2}\,dx$ |
| Work (6.5) | small displacement | $F(x)\,\Delta x$ | $\int_a^bF(x)\,dx$ |
- find the area between two curves, splitting at crossing points and choosing between vertical ($dx$) and horizontal ($dy$) strips;
- compute volumes of solids with known cross-sections, and of solids of revolution by the disk, washer and cylindrical-shell methods, about any horizontal or vertical axis;
- find the length of a curve and the area of a surface of revolution;
- compute the work done by a variable force, in springs, cables and pumping problems, and locate centres of mass of thin plates;
- set up a new application yourself by asking: what is the small piece, and what is its size?
Sections in this chapter
Areas between Curves
Top minus bottom, intersection points, curves that cross, horizontal strips and consumer surplus.
Volumes by Slicing: Disks and Washers
Known cross-sections, then solids of revolution about any horizontal or vertical line.
Volumes by Cylindrical Shells
Slice parallel to the axis instead โ often one easy integral where washers need two.
Arc Length and Surface Area
Approximate a curve by tiny straight segments; spin them to get bands of a surface.
Work, Moments and Centres of Mass
Variable forces, springs, pumping liquids, and the balance point of a thin plate.