Example 1 · level 1
Find the area (in square units) of the region bounded by y = 3x², the x-axis and the lines x = 2 and x = 3.
Show solution
- Area = [3x³/3] from 2 to 3 = 3(27 − 8)/3 = 19 sq units.
Answer: 19
CLASS 12 MATHS · APPLICATION OF INTEGRALS
Area under a curve and between curves
Area between y = f(x) ≥ 0 and the x-axis from a to b is ∫ₐᵇ f(x) dx. Between two curves, integrate (upper − lower) between their intersections.
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Worked examples
Example 2 · level 2
Find the area (in square units) enclosed between the parabola y = −3x² + 21x − 30 and the x-axis.
Show solution
- The curve cuts the x-axis at x = 2 and x = 5 and is above it in between.
- Area = [−x³ + (21/2)x² − 30x] from 2 to 5.
- = −25/2 − (−26) = 27/2 sq units.
Answer: 27/2
Example 3 · level 3
Find the area (in square units) of the region bounded by y = x² and the line y = 16.
Show solution
- Intersections: x = ±4.
- Area = [16x − x³/3] from −4 to 4 = 2(64 − 64/3) = 256/3 sq units.
Answer: 256/3