NCERT Solutions for Class 12 Mathematics Chapter 8: Applications of Integrals – Free PDF Download

Chapter 8 of Class 12 Maths is Applications of Integrals. It uses definite integrals to compute the area under curves, between curves, and bounded by circles/parabolas/ellipses. These Class 12 Mathematics Chapter 8 solutions are also useful as quick revision notes before exams.

Area Under a Curve

The area bounded by y=f(x), the x-axis, and the lines x=a, x=b is Area=∫_a^b|f(x)|dx (the modulus ensures a positive area even if the curve dips below the axis).

Area Between Two Curves

If f(x)≥g(x) on [a,b], the area between the curves is ∫_a^b[f(x)−g(x)]dx. Intersection points of the curves are found by solving f(x)=g(x) to determine the limits.

Area of Standard Curves

Circle x²+y²=a²: full area=πa², computed via 4∫₀^a√(a²−x²)dx for the first quadrant, multiplied by 4. Parabola y²=4ax and ellipse x²/a²+y²/b²=1 areas are computed similarly using integration with symmetry.

Sketching Regions

Before integrating, sketch the curves to correctly identify which function is on top/bottom (or left/right, for integration with respect to y) and determine the correct limits of integration.

Integration with Respect to y

When a region is more naturally described in terms of y (e.g. x=g(y)), the area formula becomes ∫_c^dx dy=∫_c^dg(y)dy.

Frequently Asked Questions

Why is the modulus used in the area formula?
To ensure area is always positive, since integrals below the x-axis are negative but represent real (positive) area.

How do you find the area between two curves?
Find intersection points as limits, then integrate (upper curve − lower curve).

What is the area enclosed by a full circle of radius a?
πa².

Written by Satish

NCERTBooks.org is an independent educational resource run by a small team focused on making official NCERT textbooks easy to find, read, and download for students, parents, and teachers across India. We are not affiliated with NCERT or the Ministry of Education -- we organise publicly available NCERT content by class and subject, verify links against official sources, and build tools (like our in-browser reader) that make studying more convenient. Every guide we publish is written and reviewed by our team based on the actual NCERT curriculum and syllabus.