Areas of regions bounded by a curve and the x-axis (or y-axis) between two lines, including circles, ellipses and parabolas, using definite integrals. Our own step-by-step solutions to Exercise 8.1 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. Area between \(y = f(x)\), the x-axis and \(x = a\), \(x = b\) is \(\int_a^b|y|\,dx\); between \(x = g(y)\), the y-axis and \(y = c\), \(y = d\) it is \(\int_c^d|x|\,dy\). Sketch first and use symmetry (an ellipse is four equal quarters). \(\int_0^a\sqrt{a^2 - x^2}\,dx = \tfrac{\pi a^2}{4}\).
What it tests. Where the curve dips below the x-axis, integrate each part separately and add the absolute values; odd and even symmetry halves the work.
Application of Integrals has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Application of Integrals · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.