Class 12 Mathematics Chapter 8 Applications of Integrals – Revision Notes

Once integration is established, Applications of Integrals puts it to work computing the area under a curve and the area enclosed between two curves, which calls for careful sketching first.

Last Updated: September 23, 2026

Area Formulas

• Under curve: ∫_a^b|f(x)|dx.
• Between curves: ∫_a^b[f(x)−g(x)]dx (upper−lower).
• Circle x²+y²=a²: area=πa².

Method

• Sketch curves, find intersection points as limits.
• Integrate w.r.t. y when region is naturally x=g(y).
• Split composite regions into simpler sub-integrals when needed.

One-Line Summary

Applications of Integrals uses definite integrals to compute areas under and between curves, requiring careful sketching to identify limits and which curve bounds the region above/below.

Quick visual: a worked diagram from the full Solutions page, for reference.

Frequently Asked Questions

How are definite integrals used to find the area under a curve?
The definite integral of a function between two limits gives the exact area bounded by the curve, the x-axis, and the two vertical lines at those limits, treating the region as an infinite sum of infinitesimally thin rectangles.

How is the area between two curves calculated using integration?
The area between two curves is found by integrating the difference between the upper curve and the lower curve over the interval where they bound a region, ensuring the correct curve is subtracted based on which lies above.

Written by Satish

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