A= _ab(upper-lower),dx for a region with fixed boundary order. Each vertical strip has height equal to the difference of the two ordinates; the axis is irrelevant unless it is a boundary. Find intersections; use the relevant limits; test an interior point to identify upper and lower; integrate their difference. Split only where their order changes. y=9-x2, y=2x+6 x=-3,1 and A= _-31(3-2x-x2),dx=(32)/(3). Exam check: integrating the curve alone includes the area under the line. Crossing the axis alone does not change upper minus lower.

\(A=\int_a^b(\text{upper}-\text{lower})\,\mathrm{d}x\) for a region with fixed boundary order.

Each vertical strip has height equal to the difference of the two ordinates; the axis is irrelevant unless it is a boundary.

Find intersections; use the relevant limits; test an interior point to identify upper and lower; integrate their difference. Split only where their order changes.

\(y=9-x^2,\ y=2x+6\quad\Rightarrow\quad x=-3,1\) and \(A=\int_{-3}^1(3-2x-x^2)\,\mathrm{d}x=\frac{32}{3}\).

Exam check: integrating the curve alone includes the area under the line. Crossing the axis alone does not change upper minus lower.

Line and curve boundaries

Line and curve boundaries