A= _cd(right-left),dy when both horizontal boundaries are expressed as functions of y. A horizontal strip has thickness dy and width right-hand x minus left-hand x. It is the same area principle in the other direction. Write each boundary as x=g(y); use intersection y-values as limits; test right minus left; split if the order changes. x=9-y2, x=0 y=-3,3 and A= _-33(9-y2),dy=36. Exam check: do not put x-coordinates into a dy integral. A curve left of the vertical axis needs a sign change for positive area.

\(A=\int_c^d(\text{right}-\text{left})\,\mathrm{d}y\) when both horizontal boundaries are expressed as functions of \(y\).

A horizontal strip has thickness \(\mathrm{d}y\) and width right-hand \(x\) minus left-hand \(x\). It is the same area principle in the other direction.

Write each boundary as \(x=g(y)\); use intersection \(y\)-values as limits; test right minus left; split if the order changes.

\(x=9-y^2,\ x=0\quad\Rightarrow\quad y=-3,3\) and \(A=\int_{-3}^3(9-y^2)\,\mathrm{d}y=36\).

Exam check: do not put \(x\)-coordinates into a \(\mathrm{d}y\) integral. A curve left of the vertical axis needs a sign change for positive area.

Horizontal strips and y limits

Horizontal strips and y limits