Class CBSE Class 12 Mathematics Applications of Integrals Q #874
COMPETENCY BASED
APPLY
5 Marks 2023 LA
Find the area of the region bounded by the curves x^{2}=y, y=x+2 and x-axis, using integration.
Prev Next

AI Tutor Explanation

Powered by Gemini

Step-by-Step Solution

  1. First, find the points of intersection between the curves x^{2}=y and y=x+2. Set x^{2} = x+2, which gives x^{2} - x - 2 = 0. Factoring, we get (x-2)(x+1) = 0. Thus, x = 2 or x = -1. The corresponding y values are y = 4 and y = 1. So, the points of intersection are (2, 4) and (-1, 1).
  2. Next, we need to find the area bounded by the curves and the x-axis. The region is divided into two parts. From x = -1 to x = 0, the area is under the line y = x+2. From x = 0 to x = 2, the area is bounded by the line y = x+2 above and the parabola y = x^{2} below.
  3. Calculate the area from x = -1 to x = 0: Area_1 = \int_{-1}^{0} (x+2) dx = [\frac{x^{2}}{2} + 2x]_{-1}^{0} = (0) - (\frac{1}{2} - 2) = 2 - \frac{1}{2} = \frac{3}{2}.
  4. Calculate the area from x = 0 to x = 2: Area_2 = \int_{0}^{2} (x+2 - x^{2}) dx = [\frac{x^{2}}{2} + 2x - \frac{x^{3}}{3}]_{0}^{2} = (\frac{4}{2} + 4 - \frac{8}{3}) - (0) = 2 + 4 - \frac{8}{3} = 6 - \frac{8}{3} = \frac{18 - 8}{3} = \frac{10}{3}.
  5. The total area is the sum of Area_1 and Area_2: Total Area = Area_1 + Area_2 = \frac{3}{2} + \frac{10}{3} = \frac{9 + 20}{6} = \frac{29}{6}.

Correct Answer: 29/6

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply the concepts of integration to find the area bounded by curves, which is a direct application of learned procedures and formulas.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure involving setting up integrals, finding intersection points, and evaluating the integrals to calculate the area.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It assesses the student's ability to apply integration techniques to solve a problem, rather than just recalling a formula from the textbook.

More from this Chapter

LA
33. Using integration, find the area of the region bounded by the parabola $y^{2}=4ax$ and its latus rectum.
MCQ_SINGLE
Area of the region bounded by curve \(y^{2}=4x\) and the X-axis between \(x=0\) and \(x=1\) is:
MCQ_SINGLE
The area of the region enclosed between the curve \(y=x|x|\), x-axis, \(x=-2\) and \(x=2\) is:
LA
The area of the region bounded by the line \(y=mx (m>0)\), the curve \(x^{2}+y^{2}=4\) and the \(x\)-axis in the first quadrant is \(\frac{\pi}{2}\) units. Using integration, find the value of m.
MCQ_SINGLE
The area of the region enclosed by the curve \(y=\sqrt{x}\) and the lines \(x=0\) and \(x=4\) and x-axis is:
View All Questions