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JEE Main 20146 April
Exam
JEE Main
Year
2014
Date
6 April
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2014 · 6 April · Mathematics · Area Under Curves

The area (in sq. unit) of the region described by A = \ (x, y): x ^...

The area (in sq. unit) of the region described by A={(x,y):x2+y2≤1andy2≤1−x}A = \left\{\left(x, y\right): {x}^{2} + {y}^{2} \le 1and{y}^{2} \le 1 - x\right\} isThe area (in sq. unit) of the region described by A = \ (x, y): x ^ 2 + y ^ 2 ≤ 1and y ^ 2 ≤ 1 - x\ is

Options

  1. Aπ2−23\dfrac{\pi}{2} - \dfrac{2}{3}
  2. Bπ2+23\dfrac{\pi}{2} + \dfrac{2}{3}
  3. Cπ2+43\dfrac{\pi}{2} + \dfrac{4}{3}
  4. Dπ2−43\dfrac{\pi}{2} - \dfrac{4}{3}

Correct answer

C. π2+43\dfrac{\pi}{2} + \dfrac{4}{3}

Step-by-step solution

The area (in sq. unit) of the region described by A = \ (x, y): x ^ 2 + y ^ 2 ≤ 1and y ^ 2 ≤ 1 - x\ is
Area of shaded region = area of semicircle ++ area of region BCABCA

=π2+2∫011−x⋅dx= \dfrac{\pi}{2} + 2\mathop{∫}_{0}^{1}\sqrt{1 - x} \cdot dx

=π2+2(−23)[(1−x)3/2]01= \dfrac{\pi}{2} + 2\left(- \dfrac{2}{3}\right){\left[{\left(1 - x\right)}^{3 / 2}\right]}_{0}^{1}

=π2−43(−1)= \dfrac{\pi}{2} - \dfrac{4}{3}\left(- 1\right)

=π2+43= \dfrac{\pi}{2} + \dfrac{4}{3} sq. unit

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