Application of Integrals

Complete Practice Guide: Application of Integrals Class 12

Application of Integrals is an important Class 12 Mathematics chapter because definite integrals can be interpreted geometrically and used to calculate areas of regions bounded by curves and lines. This worksheet focuses on the practical use of integration rather than only evaluating antiderivatives. The NCERT page supplied for this chapter is the official NCERT textbook portal, which identifies itself as “Textbooks PDF (I-XII)”. citeturn0view0

1. Definite Integrals and Area

If a continuous function y=f(x) is non-negative on [a,b], the area between the curve and the x-axis is Area = ∫ₐᵇ f(x) dx. If the graph lies below the x-axis, the geometric area is positive even though the definite integral is negative. Therefore, determine the position of the curve before writing the final area.

2. Area Between Two Curves

When an upper curve y=f(x) and a lower curve y=g(x) bound a region, the area is Area = ∫ₐᵇ [f(x)−g(x)] dx, provided f(x)≥g(x) throughout the interval. First find the intersection points by solving f(x)=g(x). These intersection values normally supply the limits of integration.

3. Area with the x-axis

For a curve crossing the x-axis, split the interval at the x-intercepts. The signed integral can contain positive and negative portions that cancel, whereas geometric area requires the magnitudes of the separate regions. A reliable method is to identify the sign of f(x) on each subinterval before integrating.

4. Area with the y-axis

Sometimes the curve is more conveniently expressed as x=g(y). In that case, the horizontal-strip formula is Area = ∫₍c₎⁽d⁾ [right x − left x] dy. Decide whether vertical or horizontal strips produce the simpler integral. This choice can substantially reduce algebra.

5. Finding Intersection Points

For two curves, intersection points satisfy both equations simultaneously. Equating the expressions for y often gives an equation in x. The roots that actually lie in the required region become the integration limits. Always check the ordering of the intersection points before evaluating a definite integral.

6. Symmetry

Symmetry can make area calculations much shorter. For an even function, ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx. For an odd function, ∫₋ₐᵃ f(x)dx = 0. Remember that zero signed integral does not necessarily mean zero geometric area when the question asks for total area.

7. Parabolic Regions

Many textbook-style area questions involve parabolas and straight lines. Sketch the curves, find their intersections and identify which curve is above the other. For a simple region bounded by y=x² and a horizontal or vertical line, the correct limits can often be seen immediately after solving the intersection equation.

8. Geometric Interpretation

A definite integral represents accumulated signed area. Its geometric interpretation depends on the graph and limits. The most common error is to evaluate an integral correctly but report the signed result when the question asks for geometric area. Make a quick sketch and label the region before calculating.

9. Bounded Regions

A region is bounded when the specified curves enclose a finite area. To solve such problems, identify all boundaries, locate their intersections and choose a variable for integration. If the top-minus-bottom expression changes sign, split the integral at the point where the curves meet or exchange order.

10. Exam Strategy

Use this sequence: draw → intersect → identify upper/lower or right/left → choose limits → integrate → check the sign and units. This method is useful for both straightforward questions and harder mixed applications. If the diagram is symmetric, use symmetry before performing repeated calculations.

11. Quick Formula Revision

• Area under y=f(x): ∫ₐᵇ f(x)dx
• Area between curves: ∫ₐᵇ [upper−lower]dx
• Using y: ∫ [right−left]dy
• Even function: ∫₋ₐᵃf=2∫₀ᵃf
• Odd function: ∫₋ₐᵃf=0
• Area is non-negative

12. How to Use This Worksheet

Select the topic, difficulty and question type, then press Start New Test. Attempt each question before checking it. Correct answers turn green and incorrect answers turn red. The Show Answer control remains locked until the question is checked. Use Shuffle Whole Worksheet for a new order, Timer for exam-style practice, and Save as PDF for offline revision. The mobile-first interface is designed for phones, tablets and computers.