Regions can also be measured against the \(y\)-axis. The roles of \(x\) and \(y\) swap: make \(x\) the subject, and integrate with respect to \(y\) between \(y\)-limits.
Right of the axis: \(A=\displaystyle\int_c^d x\,\mathrm{d}y\)
Left of the axis: \(\displaystyle\int_c^d x\,\mathrm{d}y\) is negative, so \(A=-\displaystyle\int_c^d x\,\mathrm{d}y\)
Matching the axis to the variable
| Boundary axis | Integral | Limits |
|---|---|---|
| \(x\)-axis | \(\displaystyle\int y\,\mathrm{d}x\) with \(y\) in terms of \(x\) | \(x\)-values |
| \(y\)-axis | \(\displaystyle\int x\,\mathrm{d}y\) with \(x\) in terms of \(y\) | \(y\)-values |
Integrating with respect to \(y\) needs everything in \(y\): make \(x\) the subject before integrating, and take the limits from the \(y\)-axis, not the \(x\)-axis.
For total area, split the vertical interval wherever \(g(y)\) changes sign and add the magnitudes. More generally, integrate right boundary minus left boundary on intervals where their ordering is fixed.
Need help? Join our JC Math tuition classes.
Learn more