Accumulate area and volume

For area between curves, integrate upper minus lower and split at crossings. A solid rotated around an axis can be sliced into disks or washers; subtract the inner radius squared from the outer radius squared.

Simpson’s rule approximates an integral using quadratic arcs. Use equally spaced samples and an even number of subintervals. The weights alternate 4 and 2 internally, with weight 1 at both ends.

\displaystyle V=\pi\int_a^b(R^2-r^2)\,dx,\quad I\approx\frac h3(y_0+4y_1+2y_2+\cdots+4y_{n-1}+y_n)

WORKED EXAMPLE

Rotate the region under y = x on 0 ≤ x ≤ 2 about the x-axis.

The disk radius is x.
Integrate πx² from 0 to 2.
The volume is 8π/3 cubic units.

Assumed knowledge

Evaluate definite integrals and find where graphs intersect. Recall signed area, circle area, volumes of elementary solids, and substitution into formulas with equally spaced data.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Accumulate area and volume

Rotate the region under y = x on 0 ≤ x ≤ 2 about the x-axis.

Put the idea to work.