Use the sketch to split the shaded region into curved and straight-edged pieces, then add or subtract their areas. The geometric subtraction formulas below apply when both boundaries are on or above the horizontal axis and the curve is above the line. Otherwise integrate upper minus lower directly, splitting wherever the boundary order changes. Horizontal line: A= _ab y,dx-area of rectangle Sloping line: A= _ab y,dx-area of triangle (or trapezium) **Shape** | **Area** Rectangle | length x breadth Triangle | 12 x base x height Trapezium | 12(a+b)h Integrating a straight-line boundary is equally valid. Use perpendicular heights in the geometric formulae.

Use the sketch to split the shaded region into curved and straight-edged pieces, then add or subtract their areas.

The geometric subtraction formulas below apply when both boundaries are on or above the horizontal axis and the curve is above the line. Otherwise integrate upper minus lower directly, splitting wherever the boundary order changes.

\(\dfrac{1}{2}\times\text{base}\times\text{height}\)

Integrating a straight-line boundary is equally valid. Use perpendicular heights in the geometric formulae.

Combining Integrals with Rectangles, Triangles and Trapeziums

Combining Integrals with Rectangles, Triangles and Trapeziums