The area bounded by the curve y^ 2 = 1 - x and the lines y = (|x|)/(x), x = - 1 and x = (1)/(2) is

Correct answer: A. ( 3 2 - (11)/(6))sq . units From the diagram, the area of the shaded region,& A = ∫ _ - 1 ^ 0 (- 1 - (- 1 - x ))dx + ∫ _ 0 ^ (1)/(2) (1 - 1 - x )dx = [- x - (1 - x) ^ (3)/(2) (3)/(2) ] _ - 1 ^ 0 + [x + (1 - x) ^ (3)/(2) (3)/(2) ] _ 0 ^ (1)/(2) = [- (2)/(3) - (1 - 2 × 2 ^ (3)/(2) 3 )] + [(1)/(2) + 2 3 × 2 ^ (3)/(2) - (2)/(3)] = 2 3 × 2 ^ (3)/(2) + 2 × 2 ^ (3)/(2) 3 - (4)/(3) - (1)/(2) = 3 2 - (4)/(3) - (1)/(2) = 3 2 - (11)/(6) sq . units

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Area Under Curves

NTA Abhyas 2020 · Mathematics · Area Under Curves

The area bounded by the curve y^ 2 = 1 - x and the lines y =...

The area bounded by the curve

y^{2} = 1 - x

and the lines

y = \dfrac{\left|x\right|}{x}, x = - 1

and

x = \dfrac{1}{2}

The area bounded by the curve y^ 2 = 1 - x and the lines y = (|x|)/(x), x = - 1 and x = (1)/(2) is

Options

A $\left(\dfrac{3}{\sqrt{2}} - \dfrac{11}{6}\right)sq . units$
B $\left(3\sqrt{2} - \dfrac{11}{4}\right)sq . units$
C $\left(\dfrac{6}{\sqrt{2}} - \dfrac{11}{5}\right)sq . units$
DNone of these

Correct answer

A. $\left(\dfrac{3}{\sqrt{2}} - \dfrac{11}{6}\right)sq . units$

Step-by-step solution

From the diagram, the area of the shaded region,&

A = \mathop{∫}_{- 1}^{0}\left(- 1 - \left(- \sqrt{1 - x}\right)\right)dx + \mathop{∫}_{0}^{\dfrac{1}{2}}\left(1 - \sqrt{1 - x}\right)dx

= {\left[- x - \dfrac{{\left(1 - x\right)}^{\dfrac{3}{2}}}{\dfrac{3}{2}}\right]}_{- 1}^{0} + {\left[x + \dfrac{{\left(1 - x\right)}^{\dfrac{3}{2}}}{\dfrac{3}{2}}\right]}_{0}^{\dfrac{1}{2}}

= \left[- \dfrac{2}{3} - \left(1 - \dfrac{2 \times {2}^{\dfrac{3}{2}}}{3}\right)\right] + \left[\dfrac{1}{2} + \dfrac{2}{3 \times {2}^{\dfrac{3}{2}}} - \dfrac{2}{3}\right]<br /><br />=\dfrac{2}{3 \times {2}^{\dfrac{3}{2}}} + \dfrac{2 \times {2}^{\dfrac{3}{2}}}{3} - \dfrac{4}{3} - \dfrac{1}{2}<br /><br />=\dfrac{3}{\sqrt{2}} - \dfrac{4}{3} - \dfrac{1}{2}<br /><br />=\dfrac{3}{\sqrt{2}} - \dfrac{11}{6}

sq . units

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