Chapter Hub
Application of Integrals
Application of Integrals uses definite integrals to find areas of plane regions. The central idea is to slice a region into thin strips, express each strip as height times small width, and add all strips through integration. For regions under one curve, the area is usually written with respect to the x-axis as ∫ y dx. The limits must match the part of the curve that actually bounds the required region. For standard curves such as circles, parabolas, ellipses and lines, the diagram decides the integral. Symmetry, correct limits, and the correct half of a curve often reduce work and prevent sign errors. For area between two curves, you must first find the intersection points and decide which curve is above the other on the interval. The required area is the integral of top curve minus bottom curve, not merely the difference of two separate-looking formulae.
Difficulty
Medium
Study time
70-90 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 70 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Key Concepts
Work through the concepts in order. Each one has an explanation, a worked example, a self-check and an exam answer.
All Concepts
All 3 concepts of the chapter, in the order the NCERT chapter teaches them.
Area Under a Curve Using Definite Integral
The area bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b is found by adding thin vertical strips from x = a to x = b. If f(x) is non-negative on [a, b], the area is ∫ab f(x) dx.
Area Bounded by Standard Curves
Area bounded by standard curves is found by first sketching the known curve shape, identifying the required bounded region, and then setting a definite integral with correct limits. Common curves include circles, parabolas, ellipses and straight lines.
Area Between Two Curves
The area between two curves y = f(x) and y = g(x) from x = a to x = b is ∫ab [upper curve - lower curve] dx, provided the same curve stays above on the whole interval.
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
High Probability Topics
- Area Under a Curve Using Definite Integral
- Area Bounded by Standard Curves
- Area Between Two Curves
Common Traps
- Using signed area as final area when the curve lies below the x-axis.
- Subtracting curves in the order given instead of upper minus lower.
- Forgetting to solve intersections before setting limits.
- Applying symmetry to a region that is not symmetric.
- Using denominator values in ellipse equation as semi-axis lengths instead of taking square roots.
- Not writing square units in the final answer.
How CBSE Asks This Chapter
- Area Under a Curve Using Definite Integral
Usually asked as a short or long answer question requiring limits from given boundaries, direct integration, and final area in square units. It may also appear with a curve and two vertical lines, where no intersection solving is needed.
- Area Bounded by Standard Curves
Questions commonly ask for area enclosed by a circle, ellipse, parabola with a line, or a region in one quadrant. Marks are awarded for a correct sketch, correct limits, justified symmetry, and valid integral setup.
- Area Between Two Curves
Often asked as a long answer problem where students must solve intersections, sketch the region, choose the correct subtraction order, integrate a polynomial or standard expression, and state the final area.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Area Under a Curve Using Definite Integral
Remember
Area = ∫ab y dx = ∫ab f(x) dx when the region is bounded by y = f(x), x-axis, x = a, x = b, and f(x) ≥ 0 on [a, b].Trap to avoid
Students often write ∫ab y dx without checking whether y is above or below the x-axis throughout the interval.
Area Bounded by Standard Curves
Remember
Useful forms and conditions: Circle x2 + y2 = a2 has upper branch y = √(a2 - x2), -a ≤ x ≤ a.Trap to avoid
A frequent error is using the full circle or full ellipse area formula when the question asks for only one bounded part.
Remember
Area = ∫ab [f(x) - g(x)] dx when f(x) ≥ g(x) on [a, b].Trap to avoid
Many students subtract in the order written in the question instead of subtracting lower curve from upper curve.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Area Under a Curve Using Definite Integral | Area = ∫ab y dx = ∫ab f(x) dx when the region is bounded by y = f(x), x-axis, x = a, x = b, and f(x) ≥ 0 on [a, b]. | Students often write ∫ab y dx without checking whether y is above or below the x-axis throughout the interval. |
| Area Bounded by Standard Curves | Useful forms and conditions: Circle x2 + y2 = a2 has upper branch y = √(a2 - x2), -a ≤ x ≤ a. | A frequent error is using the full circle or full ellipse area formula when the question asks for only one bounded part. |
| Area Between Two Curves | Area = ∫ab [f(x) - g(x)] dx when f(x) ≥ g(x) on [a, b]. | Many students subtract in the order written in the question instead of subtracting lower curve from upper curve. |
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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