Application of Integrals

part 1

Area Under Simple Curves

• Area bounded by curve and x-axis
• Area bounded by curve and y-axis
• Area using definite integrals

part 2

Area Between Curves

• Area between two curves
• Finding intersection points
• Setting up integrals for area problems

part 3

Applications and Special Cases

• Area of ellipse and parabolic regions
• Area using both x and y integration
• Mixed problems on area calculation

Area under curve (above x-axis)

$A = \int_a^b f(x)\,dx$ [f(x) ≥ 0]

★ board favourite

Area between two curves

$A = \int_a^b [f(x) - g(x)]\,dx$ [f(x) ≥ g(x)]

★ board favourite

Area w.r.t. y-axis

$A = \int_c^d g(y)\,dy$

★ frequently asked

Area of full circle x²+y²=a²

$A = \pi a^2$

★ frequently asked

Area of ellipse x²/a²+y²/b²=1

$A = \pi ab$

★ frequently asked

Using integration, find the area of the region bounded by the curve y = x² and the line y = x in the first quadrant.

Find the area of the region bounded by the parabola y² = 8x and its latus rectum, using integration.

Using integration, find the area of the region bounded by the parabola y² = 4x and the line x = 3.

Using integration, find the area of the region {(x, y) : x² + y² ≤ 4, x + y ≥ 2} in the first quadrant.

Using integration, find the area of the triangle whose vertices are (1, 0), (2, 2) and (3, 1).

Using integration, find the area of the region bounded by the line 2y = 5x + 7, the x-axis, and the lines x = 2 and x = 8.

Using integration, find the area enclosed by the circle x² + y² = a².

Write the integral that gives the area bounded by the curve y = x³, the x-axis and the ordinates x = 1 and x = 2.

Using integration, find the area of the region bounded by the ellipse x²/9 + y²/4 = 1.

How many marks is Application of Integrals worth in the Class 12 board exam?

Around 5 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Area between curves — sketch first, then integrate.

Is Application of Integrals easy or hard?

It's rated medium — a moderate-difficulty chapter in Class 12 Maths. Most students need about 3 hours to cover it well.

What are the important topics in Application of Integrals?

The chapter covers 6 NCERT topics in 3 areas: Area Under Simple Curves; Area Between Curves; Applications and Special Cases.

What questions come from Application of Integrals in board exams?

Between 2019–2024, CBSE board papers asked questions from this chapter on Area between Curves, Area under Parabola, Area under a Curve, Area of a Region — 1- to 5-mark questions.