When a curve goes below the x-axis, the integral of that part is negative. To find a total area, split the interval at each root, integrate each part separately and add the sizes (ignore the minus signs).
For y = x² − 5x + 4 from x = 1 to x = 5, the curve crosses the axis at x = 1 and x = 4. The part from 1 to 4 is below the axis and the part from 4 to 5 is above. Integrating straight from 1 to 5 would let the two parts cancel and give the wrong total.
Worked example
An integral from 0 to 2 gives −3 and from 2 to 4 gives 5. What is the total area?
- The first region is below the axis: its area is 3.
- Add the sizes: 3 + 5.
Answer:8 square units
Key words
- signed area
- area counted negative below the axis
- total area
- the sum of the sizes of all regions
- split
- integrate each region separately
Practise this
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