Using integration find the area of the region \(x,y): x²-4y≤0, y-x≤0\

MathsCBSE Class 12 · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2025-26: question 28(B)

From the official CBSE sample question paper for 2025-26, section C, question 28(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(B) · 3 marks · Short answerApplication of Integrals

Using integration find the area of the region ${(x, y) : x^{2} - 4 y \leq 0, y - x \leq 0}$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Identify the two curves $x^{2} = 4 y$ and $y = x$ and find where they meet.

Hint 2

Area = $\int$ (upper curve − lower curve) $d x$ between the intersection points.

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.

The region lies above the parabola $x^{2} = 4 y$ (i.e. $y \geq x^{2} /4$ ) and below the line $y = x$ . Solving $\frac{x ^{2}}{4} = x$ gives $x = 0, 4$ , so they meet at $(0, 0)$ and $(4, 4)$ . Sketch: upward parabola with vertex O and the line $y = x$ ; shade the region between them from $x = 0$ to $x = 4$ .
1 mark
Required area $= \int_{0}^{4} x d x - \int_{0}^{4} \frac{x ^{2}}{4} d x$ .
1 mark
$= [\frac{x ^{2}}{2}]_{0}^{4} - \frac{1}{12} [x^{3}]_{0}^{4}$ .
½ mark
$= 8 - \frac{64}{12} = 8 - \frac{16}{3} = \frac{8}{3}$ sq. units.
½ mark

Answer: $\frac{8}{3}$ sq. units

Where the marks are

Finds intersection points and sketches/shades region (1 mark)
Sets up area as $\int_{0}^{4} (x - x^{2} /4) d x$ (1 mark)
Integrates correctly (½ mark)
Area $8/3$ sq. units (½ mark)

Revise the chapter

Chapter: Application of Integrals (notes, formulas, where marks are lost and every past question)
Topic: Area Involving a Parabola

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 paperSample paper 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 28(B)

Hints

Step-by-step solution

Revise the chapter

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