Application of Integrals – Class 12 Mathematics notes, important questions and MCQs

Chapter overview

Definite integrals find areas under curves and between curves.

Area under a curve

• Area between y = f(x), the x-axis, x = a and x = b: ∫ₐᵇ |f(x)| dx.
• Area with the y-axis: ∫ x dy between y-limits.
• If the curve goes below the x-axis, take the absolute value of that part.

Standard results

• Area of the circle x² + y² = a² is πa² (4 × quadrant area).
• Area of the ellipse x²/a² + y²/b² = 1 is πab.
• Area between two curves = ∫(upper − lower) dx between intersection points.

Key terms

Area under a curve

region bounded by the curve, the axis and two ordinates

Area between two curves

∫ (upper curve − lower curve) dx between their points of intersection

Area along the y-axis

∫ x dy between y-limits, used when the region is easier to describe horizontally

Formulas

• Area = ∫ₐᵇ (f(x) − g(x)) dx, f ≥ g
• Ellipse area = πab

Important questions with answers

Quick revision

• Use symmetry to reduce work.
• Area is always positive.
• Circle area πa², ellipse πab.

Exam tips

• Sketch the region first to find limits and which curve is on top.
• Find points of intersection first; if a curve crosses the x-axis, split the integral and add absolute values.

Practice MCQs

1. Area of the ellipse x²/9 + y²/4 = 1 is

   1. 13π
   2. 6π
   3. 36π
   4. 12π
   Show answer

   Answer: (b) 6π. π × 3 × 2.

2. Area under y = x from x = 0 to x = 2 is

   1. 1
   2. 2
   3. 4
   4. 8
   Show answer

   Answer: (b) 2. ½ × 2 × 2.

3. Area bounded by y = x² , the x-axis and x = 3 is

   1. 27
   2. 9
   3. 3
   4. 6
   Show answer

   Answer: (b) 9. [x³/3]₀³.

4. Area of a quadrant of x² + y² = 16 is

   1. 8π
   2. 16π
   3. 4π
   4. 2π
   Show answer

   Answer: (c) 4π. ¼ × 16π.

5. Area bounded by y = 2x, the x-axis and x = 3 is

   1. 6
   2. 9
   3. 18
   4. 3
   Show answer

   Answer: (b) 9. ½ × 3 × 6.

6. Area between y = x² and y = x is

   1. 1/6
   2. 1/3
   3. 1/2
   4. 1/12
   Show answer

   Answer: (a) 1/6. ∫₀¹ (x − x²) dx.

7. Area under y = sin x from 0 to π is

   1. 2
   2. 1
   3. 0
   4. π
   Show answer

   Answer: (a) 2. [−cos x]₀^π = 2.

8. Area enclosed by the circle x² + y² = 4 is

   1. 2π
   2. 4π
   3. 8π
   4. 16π
   Show answer

   Answer: (b) 4π. πr² with r = 2.