When the boundary is simpler as a function of height, use horizontal strips. Write the boundaries as \(x=\mathrm{g}(y)\); the limits are vertical coordinates.
\(A=\int_c^d|\mathrm{g}(y)|\,\mathrm{d}y\) for a region between the curve and the vertical axis.
Between two curves written as horizontal coordinates \(x=\mathrm{R}(y)\) and \(x=\mathrm{L}(y)\), use \(A=\int_c^d[\mathrm{R}(y)-\mathrm{L}(y)]\,\mathrm{d}y\), with right minus left.
For \(x=y^2\), the vertical axis and \(y=2\), the lower boundary is \(y=0\):
\(A=\int_0^2y^2\,\mathrm{d}y=\left[\dfrac{y^3}{3}\right]_0^2=\dfrac83\).
When solving for \(x\), choose the branch that actually bounds the region. Split the integral if that boundary changes.
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