Application of Integrals - BSEB Inter Study Guide
Application of Integrals
Questions ask for the area bounded by curves, axes and lines. The exam tests setting up the correct definite integral, especially deciding which curve is upper and where the curves intersect. A rough sketch is part of the solution, not decoration.
Core rules
- Area between y=f(x) and the x-axis from x=a to b is ∫ from a to b of |f(x)|dx.
- Area between upper curve y=f(x) and lower curve y=g(x) is ∫ from a to b of [f(x)−g(x)]dx.
- Find intersection points by solving f(x)=g(x); these become limits.
- For curves of the form x=φ(y), integrate with respect to y: ∫(right−left)dy.
- Use symmetry only when the region is genuinely symmetric about an axis.
- If vertical lines x=a and x=b bound the region, use them as limits after confirming the upper-lower order.
Worked example
Find the area bounded by y=x² and y=x. Intersections: x²=x gives x=0,1. The curves meet at (0,0) and (1,1). On (0,1), x>x², so area=∫ from 0 to 1 of (x−x²)dx=[x²/2−x³/3] from 0 to 1=1/2−1/3=1/6 square unit.
Traps
- Subtracting in the wrong order and reporting a negative area.
- Using x=0 and x=1 without checking which curve is above.
- Forgetting to split the integral when curves cross inside the interval.
- Integrating y=x² and y=x separately and adding, which counts only the region under each, not the bounded region.
- Reporting a negative value as an area.