Make a rough sketch of the region \(x,y):0≤ y≤ x²+1,\ 0≤ y≤ x+1,\ 0≤ x≤2\ and find the area of the region, using the method of integration.

MathsCBSE Class 12 · Sample paper, session 2023-24 · 5 marks · Long answer

CBSE Class 12 Mathematics sample paper 2023-24: question 32

From the official CBSE sample question paper for 2023-24, section D, question 32, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32 · 5 marks · Long answerApplication of Integrals

Make a rough sketch of the region ${(x, y) : 0 \leq y \leq x^{2} + 1, 0 \leq y \leq x + 1, 0 \leq x \leq 2}$ and find the area of the region, using the method of integration.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

Find where $y = x^{2} + 1$ meets $y = x + 1$ .

Hint 2

On each part of $[0, 2]$ , the upper boundary is whichever curve is lower. Split the integral at the intersection point.

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.

Sketch: draw the parabola $y = x^{2} + 1$ (vertex $(0, 1)$ ) and the line $y = x + 1$ for $0 \leq x \leq 2$ . The region lies above the $x$ -axis and below the lower of the two curves. It is bounded by $P (0, 1)$ , the parabola up to $Q (1, 2)$ , the line from $Q (1, 2)$ to $R (2, 3)$ , the line $x = 2$ down to $T (2, 0)$ , and the $x$ -axis back to $O$ .
1 mark
Intersection: $x^{2} + 1 = x + 1 \Rightarrow x (x - 1) = 0 \Rightarrow x = 0, 1$ , giving $P (0, 1)$ and $Q (1, 2)$ . On $[0, 1]$ , $x^{2} + 1 \leq x + 1$ ; on $[1, 2]$ , $x + 1 \leq x^{2} + 1$ .
1 mark
Required area $= \int_{0}^{1} (x^{2} + 1) d x + \int_{1}^{2} (x + 1) d x$ .
1 mark
$= [\frac{x ^{3}}{3} + x]_{0}^{1} + [\frac{x ^{2}}{2} + x]_{1}^{2}$ .
½ mark
$= (\frac{1}{3} + 1) + [(2 + 2) - (\frac{1}{2} + 1)] = \frac{4}{3} + \frac{5}{2}$ .
½ mark
$= \frac{23}{6}$ sq units.
1 mark

Answer: $\frac{23}{6}$ sq units

Where the marks are

Correct rough sketch of the region (1 mark)
Finds the points of intersection $(0, 1)$ and $(1, 2)$ (1 mark)
Sets up the area as the sum of two integrals (1 mark)
Integrates correctly (½ mark)
Substitutes the limits correctly (½ mark)
Final area $\frac{23}{6}$ sq units (1 mark)

Revise the chapter

Chapter: Application of Integrals (notes, formulas, where marks are lost and every past question)

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 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 32

Hints

Step-by-step solution

Revise the chapter

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