Find the area of the region bounded by the curve 4x²+y²=36 using integration.

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

CBSE Class 12 Mathematics 2024: question 33

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerApplication of Integrals

Find the area of the region bounded by the curve $4 x^{2} + y^{2} = 36$ using integration.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

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

Hint 1

Divide by 36 to write the ellipse in standard form and find a and b.

Hint 2

Use symmetry: total area is 4 times the first-quadrant area.

Hint 3

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

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.

$4 x^{2} + y^{2} = 36 \Rightarrow \frac{x ^{2}}{9} + \frac{y ^{2}}{36} = 1$ , an ellipse centred at the origin with vertices $(\pm 3, 0)$ and $(0, \pm 6)$ . Figure: ellipse elongated along the y-axis, symmetric about both axes; the first-quadrant part (from $x = 0$ to $x = 3$ ) is shaded.
1 mark
By symmetry, Area $= 4 \int_{0}^{3} y d x$ with $y = 2 9 - x^{2}$ (first quadrant), so Area $= 4 \int_{0}^{3} 2 9 - x^{2} d x = 8 \int_{0}^{3} 9 - x^{2} d x$ .
1½ marks
$= 8 [\frac{x}{2} 9 - x^{2} + \frac{9}{2} sin^{- 1} \frac{x}{3}]_{0}^{3} = 8 [0 + \frac{9}{2} \cdot \frac{π}{2} - 0]$
1½ marks
$= 18 π$ sq. units.
1 mark

Answer: $18 π$ sq. units

Where the marks are

Standard form of ellipse and correct sketch (1 mark)
Sets up Area $= 4 \int_{0}^{3} 2 9 - x^{2} d x$ (1½ marks)
Correct integration with limits (1½ marks)
Area $= 18 π$ sq. units (1 mark)

Revise the chapter

Chapter: Application of Integrals (notes, formulas, where marks are lost and every past question)
Topic: Area of Circle and Ellipse by Integration

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 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 33

Hints

Step-by-step solution

Revise the chapter

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