CBSE Math RevisionApplication of Integrals: CBSE Class 12 Maths knowledge organiser
Everything to know about application of integrals on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.
Key definitions
- Area under a curve
- The area between y = f(x), the x-axis and two vertical lines: ∫ₐᵇ y dx.
- Vertical strip
- A thin rectangle of width dx used to build up the area.
- Region
- The part of the plane bounded by the given curves and lines; sketch it first.
Key formulas
| Area | \(\int_a^b|y|\,dx,\) \(\int_c^d|x|\,dy\) |
| Circle \(x^2+y^2=a^2\): area \(\pi a^2\); ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\): area \(\pi ab\) | |
| Region between a curve and a line: integrate (upper − lower) between the intersection points |
Worked example
Area bounded by \(y^2 = 4x\) and \(x = 1\): by symmetry, \(2\displaystyle\int_0^1 2\sqrt x\,dx = \dfrac83\) sq units.
Common mistakes
- Reporting a negative or zero “area” because part of the curve is below the axis.
- Forgetting to double when using symmetry — or doubling a region that is not symmetric.
- Wrong limits: using the intercepts of the line instead of the intersection points with the curve.
- Integrating \(y\) with respect to \(y\) (mixing up vertical and horizontal strips).
You should be able to…
- Find the area under a curve with vertical strips and the areas of circles and ellipses using symmetry.
- Choose vertical or horizontal strips and find the area between a curve and a line from their intersection points.
- Write full-marks 5-mark area answers and case studies: sketch, shading, limits, integral and sq units.
- Handle regions that need two integrals, parts of circles and ellipses and area questions where symmetry is only partial.
The printable sheet
Revise it next
- Application of Integrals: Class 12 notes
- Practise Application of Integrals by step (Route to 95)
- Skill Builders
- CBSE Class 12 Maths formula sheet (PDF)
Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers