Using integration, find the area of the ellipse (x²)/(16) + (y²)/(4) = 1, included between the lines x = -2 and x = 2.

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2024: question 34

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section D, question 34, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34 · 5 marks · Long answerApplication of Integrals

Using integration, find the area of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{4} = 1$ , included between the lines $x = - 2$ and $x = 2$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Solve the ellipse equation for $y$ and sketch the region between $x = - 2$ and $x = 2$ .

Hint 2

The region is symmetric about both axes, so find the area in the first quadrant from $x = 0$ to $x = 2$ and multiply by 4.

Hint 3

Use $\int a^{2} - x^{2} d x = \frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a}$ with $a = 4$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Figure: draw the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{4} = 1$ centred at the origin, with vertices $(\pm 4, 0)$ and $(0, \pm 2)$ . Draw the vertical lines $x = - 2$ and $x = 2$ and shade the part of the ellipse between them.
1 mark
From the equation, $y = \frac{1}{2} 16 - x^{2}$ for the upper half. The region is symmetric about both axes, so
Area $= 4 \int_{0}^{2} y d x = 4 \cdot \frac{1}{2} \int_{0}^{2} 4^{2} - x^{2} d x$
1 mark
Using $\int a^{2} - x^{2} d x = \frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a}$ :
Area $= 2 [\frac{x}{2} 16 - x^{2} + 8 sin^{- 1} \frac{x}{4}]_{0}^{2}$
2 marks
$= 2 [12 + 8 \cdot \frac{π}{6}] = 2 [23 + \frac{4 π}{3}] = 43 + \frac{8 π}{3}$ sq units
1 mark

Answer: $43 + \frac{8 π}{3}$ square units

Where the marks are

Correct figure of the ellipse with the region between $x = - 2$ and $x = 2$ shaded (1 mark)
Sets up the area as $4 \int_{0}^{2} \frac{1}{2} 16 - x^{2} d x$ using symmetry (1 mark)
Integrates correctly using the standard formula for $a^{2} - x^{2}$ (2 marks)
Substitutes the limits to get $43 + \frac{8 π}{3}$ (1 mark)

Revise the chapter

Chapter: Application of Integrals (notes, formulas, where marks are lost and every past question)
Topic: Area Under Lines and Modulus Graphs

Practise this chapterBoard-style questions from Application of Integrals, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2024: question 34

Hints

Step-by-step solution

Revise the chapter

More questions from Application of Integrals