The area bounded by y = f(x) (above the X-axis), the X-axis and x = a, x = b is:

∫[a, b] y dx. Sum of vertical strips of height y.

The area bounded by x = g(y), the Y-axis and the lines y = c and y = d is:

∫[c, d] x dy. Horizontal strips of length x.

Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 1 Chapter 7: Application of Definite Integration

Short notes, key concepts, quick revision and practice MCQs with answers

Definite integrals measure areas. The chapter finds the area under a curve above the X-axis, the area between a curve and the Y-axis, areas of regions lying below the X-axis and the area bounded by two curves.

Key concepts: Application of Definite Integration

Area under a curve: The area bounded by y = f(x), the X-axis and the lines x = a and x = b (curve above the X-axis) is A = ∫[a, b] y dx = ∫[a, b] f(x) dx.
Area with the Y-axis: The area bounded by x = g(y), the Y-axis and the lines y = c and y = d is A = ∫[c, d] x dy.
Region below the X-axis: If the curve lies below the X-axis, the integral is negative, so the area is |∫[a, b] f(x) dx|.
Area between two curves: If f(x) ≥ g(x) on [a, b], the area between them is ∫[a, b] [f(x) − g(x)] dx. The limits are the x-coordinates of the points of intersection.
Symmetry: For a curve symmetric about an axis (such as y² = 4ax or a circle), find the area of one part and multiply by the number of symmetric parts.
Steps: Sketch the region, find the points of intersection and limits, choose vertical (dx) or horizontal (dy) strips, integrate and take the absolute value.

Quick revision: important points

Area is always non-negative.
Area of the circle x² + y² = a² is πa².
Area of the ellipse x²/a² + y²/b² = 1 is πab.
Area bounded by y = x and y = x² is 1/6 square unit.
Area bounded by y² = 4ax and the line x = a is (8/3)a².

Practice MCQs with answers

The area bounded by y = f(x) (above the X-axis), the X-axis and x = a, x = b is:
1. ∫[a, b] x dy
2. ∫[a, b] y dx
3. ∫[a, b] (x + y) dx
4. f(b) − f(a)
Show answer
Answer: B. ∫[a, b] y dx
Sum of vertical strips of height y.
The area bounded by y = x, the X-axis and the lines x = 0 and x = 2 is:
1. 1
2. 2
3. 4
4. 8
Show answer
Answer: B. 2
∫[0, 2] x dx = 2.
The area under y = x² from x = 0 to x = 3 is:
1. 3
2. 6
3. 9
4. 27
Show answer
Answer: C. 9
[x³/3]₀³ = 9.
The area bounded by x = g(y), the Y-axis and the lines y = c and y = d is:
1. ∫[c, d] y dx
2. ∫[c, d] x dy
3. ∫[c, d] (x − y) dy
4. g(d) − g(c)
Show answer
Answer: B. ∫[c, d] x dy
Horizontal strips of length x.
The area bounded by y = 2x, the X-axis and the line x = 3 is:
1. 6
2. 9
3. 18
4. 3
Show answer
Answer: B. 9
∫[0, 3] 2x dx = 9.

All Class 12 Maths and Statistics chapters

Class 12 Maths and Statistics notes: all chapters