Express the curve as \(x=\mathrm{g}(y)\) and use \(y\)-limits, \(a<b\).

\(A=\int_a^b|\mathrm{g}(y)|\,\mathrm dy\)

\(A=\displaystyle\int_a^b \mathrm{g}(y)\,\mathrm{d}y\)

\(A=-\displaystyle\int_a^b \mathrm{g}(y)\,\mathrm{d}y=\left|\displaystyle\int_a^b \mathrm{g}(y)\,\mathrm{d}y\right|\)

\(\displaystyle\int y\,\mathrm{d}x\) with \(y\) in terms of \(x\)

\(\displaystyle\int x\,\mathrm{d}y\) with \(x\) in terms of \(y\)

Split where \(\mathrm{g}(y)\) changes sign. For two boundaries, integrate right minus left on each interval where their order is fixed.

Area Enclosed by the Curve and the \(y\)-axis