Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Areas with the y-Axis and Volumes of Revolution

Use horizontal or vertical strips to model area and volume about either coordinate axis.

AA HL · AHL 5.17

Learning goal

Set up and evaluate areas measured from the y-axis and volumes of revolution about the x- or y-axis.

Syllabus link

AA HL: AHL 5.17 · Topic 5 Calculus.

Big idea

The geometry chooses the variable: horizontal strips naturally lead to dy; vertical strips to dx.

Key relationship

\(A=\int_c^d|g(y)|dy\)

1

Core idea

The geometry chooses the variable: horizontal strips naturally lead to dy; vertical strips to dx.

Set up and evaluate areas measured from the y-axis and volumes of revolution about the x- or y-axis.

2

Key results and method

\[A=\int_c^d|g(y)|dy\]
\[V_x=\pi\int_a^b[f(x)]^2dx\]
\[V_y=\pi\int_c^d[g(y)]^2dy\]
\[V=\pi\int(R^2-r^2)\,d(\text{variable})\]
  1. Sketch the region and axis of rotation.
  2. Choose strips perpendicular to the rotation axis for disk/washer volumes.
  3. Identify outer and inner radii when a hole is present.
  4. Square the radius functions before integrating.
3

Worked example

Question

Rotate the region under \(y=x\), \(0\le x\le2\), about the x-axis.

Solution
  1. Each cross-section is a disk of radius \(y=x\).
  2. \(V=\pi\int_0^2x^2dx\).
Answer: \(V=\frac{8\pi}{3}\).
4

Practice

Try these questions before moving on.

  1. Find an area bounded by a curve and the y-axis using dy.
  2. Rotate a region about the y-axis using \(x=g(y)\).
  3. Set up a washer volume as the difference of two volumes.

Search the worksheet library for more practice →

5

Tutorials and premium resources

Browse the Radford Mathematics video library →

Premium printable lesson notes

The finalized printable handout for this topic is a premium Radford Mathematics resource and is not offered as a free website download.

Visit the Radford Mathematics Store →