Find the area of the region bounded by the curves x² = y, y = x + 2 and x-axis, using integration.

MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2023: question 35

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, section D, question 35, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35 · 5 marks · Long answerApplication of Integrals

Find the area of the region bounded by the curves $x^{2} = y$ , $y = x + 2$ and x-axis, using integration.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

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

Hint 1

Sketch all three curves and find where the line meets the parabola and where it meets the x-axis.

Hint 2

The region touches the x-axis from $x = - 2$ to $x = 0$ , but its upper boundary changes at $x = - 1$ .

Hint 3

Split the area into two integrals: the line on $[- 2, - 1]$ and the parabola on $[- 1, 0]$ .

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: the upward parabola $y = x^{2}$ has its vertex at $O$ . The line $y = x + 2$ meets the x-axis at $(- 2, 0)$ . The region bounded by the line, the parabola and the x-axis lies above the x-axis between $x = - 2$ and $x = 0$ . It is bounded by the line from $(- 2, 0)$ to $(- 1, 1)$ and by the parabola from $(- 1, 1)$ to $(0, 0)$ . The vertical line $x = - 1$ splits it into two parts.
1½ marks
Intersection of the curves: $x^{2} = x + 2 \Rightarrow x^{2} - x - 2 = 0 \Rightarrow x = - 1, 2$ . The relevant corner is $(- 1, 1)$ .
½ mark
Required area $= \int_{- 2}^{- 1} (x + 2) d x + \int_{- 1}^{0} x^{2} d x$ .
1½ marks
$= [\frac{( x + 2 ) ^{2}}{2}]_{- 2}^{- 1} + [\frac{x ^{3}}{3}]_{- 1}^{0} = (\frac{1}{2} - 0) + (0 + \frac{1}{3})$
1 mark
$= \frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ square units.
½ mark

Answer: $\frac{5}{6}$ square units

Where the marks are

Correct graph with the region shaded (1½ marks)
Points of intersection $x = - 1, 2$ (½ mark)
Correct integral set-up $\int_{- 2}^{- 1} (x + 2) d x + \int_{- 1}^{0} x^{2} d x$ (1½ marks)
Correct integration and substitution of limits (1 mark)
Area $= 5/6$ (½ 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 paper2023 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 35

Hints

Step-by-step solution

Revise the chapter

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