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NCERT Solutions · Class 12 · Chapter 8: Application of Integrals

NCERT Solutions for Class 12 Maths Chapter 8 Exercise 8.1

Exercise 8.1: Area under simple curves. 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). \(\int_0^a\sqrt{a^2 - x^2}\,dx = \tfrac{\pi a^2}{4}\).

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Exercise 8.1 questions and solutions

Exercise 8.1, Question 1

Find the area enclosed by the ellipse \(\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1\).
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  1. In the first quadrant \(y = \tfrac34\sqrt{16 - x^2}\), \(0 \le x \le 4\); the ellipse is symmetric about both axes.
  2. Area \[\begin{aligned}= 4 \cdot \tfrac34\displaystyle\int_0^4\sqrt{16 - x^2}\,dx &= 3\left[\dfrac{x}{2}\sqrt{16 - x^2} + 8\sin^{-1}\dfrac{x}{4}\right]_0^4 \\ &= 3 \cdot 8 \cdot \dfrac{\pi}{2}\end{aligned}\]
Answer: \(12\pi\) square units

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Exercise 8.1, Question 2

Find the area enclosed by the ellipse \(\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1\).
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  1. First quadrant: \(y = \tfrac32\sqrt{4 - x^2}\), \(0 \le x \le 2\).
  2. Area \[\begin{aligned}= 4 \cdot \tfrac32\displaystyle\int_0^2\sqrt{4 - x^2}\,dx &= 6\left[\dfrac{x}{2}\sqrt{4 - x^2} + 2\sin^{-1}\dfrac{x}{2}\right]_0^2 \\ &= 6 \cdot 2 \cdot \dfrac{\pi}{2}\end{aligned}\]
Answer: \(6\pi\) square units

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Exercise 8.1, Question 3

The area in the first quadrant bounded by \(x^2 + y^2 = 4\), \(x = 0\) and \(x = 2\) is: (A) \(\pi\) (B) \(\tfrac{\pi}{2}\) (C) \(\tfrac{\pi}{3}\) (D) \(\tfrac{\pi}{4}\)
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  1. A quarter of a circle of radius 2: \[\displaystyle\int_0^2\sqrt{4 - x^2}\,dx = \tfrac14\pi(2)^2\]
Answer: (A) \(\pi\)

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Exercise 8.1, Question 4

The area bounded by \(y^2 = 4x\), the y-axis and the line \(y = 3\) is: (A) 2 (B) \(\tfrac94\) (C) \(\tfrac93\) (D) \(\tfrac92\)
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  1. Use horizontal strips: \(x = \dfrac{y^2}{4}\) from \(y = 0\) to 3.
  2. \[\begin{aligned}\displaystyle\int_0^3\dfrac{y^2}{4}\,dy &= \left[\dfrac{y^3}{12}\right]_0^3 \\ &= \dfrac{27}{12}\end{aligned}\]
Answer: (B) \(\tfrac94\)

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Done the NCERT exercises? The board paper asks more

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.

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Application of Integrals in our sample papers: Sample paper 1 (question 33) · Sample paper 2 (question 29) · Sample paper 3 (question 33) · Sample paper 4 (question 33).

Also useful: free MCQs and case studies for Application of Integrals · formulas for this chapter (Class 12 formula sheet, free PDF) · 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.