The area between a curve, the x-axis and the lines x = a and x = b is ∫ₐᵇ y dx, when the curve is above the x-axis.
For the area enclosed by y = −x² − 2x + 3 and the x-axis, first find where the curve crosses the axis: −(x + 3)(x − 1) = 0, so x = −3 and x = 1. Then integrate between these limits: [−x³/3 − x² + 3x] from −3 to 1 = 5/3 − (−9) = 32/3.
Worked example
Find the area under y = x + 1 from x = 1 to x = 3.
- ∫₁³ (x + 1) dx = [x²/2 + x]₁³.
- (4.5 + 3) − (0.5 + 1) = 6.
Answer:6 square units
Key words
- area under a curve
- between the curve and the x-axis
- roots
- where the curve meets the x-axis
- square units
- units of area
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