Mathematics - Area Under Curves Question with Solution | TestHub

MathematicsArea Under CurvesArea bounded by Miscellaneous CurvesHard2 minPYQ_2018

MathematicsHardsingle choice

The area of the region included between the curves $x^{2} + y^{2} = a^{2}$ and $\sqrt{|x|} + \sqrt{|y|} = \sqrt{a} (a > 0)$ is

Answer:

Solution:

Given curves are

$\sqrt{|x|} + \sqrt{|y|} = \sqrt{a} . . . (i)$

and $x^{2} + y^{2} = a^{2} . . . (ii)$

Now, required areas $= 4$ [shaded area in the first quadrant]

$= 4 [\frac{π a^{2}}{4} - \int_{0}^{a} {(\sqrt{a} - \sqrt{x})}^{2} d x]$ [from Eqs. (i) and (ii)]

$= 4 [\frac{π a^{2}}{4} - \int_{0}^{a} (a + x - 2 \sqrt{a} \sqrt{x}) d x]$

$= 4 \frac{π a^{2}}{4} - 4 {[a x + \frac{x^{2}}{2} - \frac{4}{3} \sqrt{a} x^{3 / 2}]}_{0}^{a}$

$= (π - \frac{2}{3}) a^{2}$ sq units

Hence, option (a) is correct.

Stream:BITSATSubject:MathematicsTopic:Area Under CurvesSubtopic:Area bounded by Miscellaneous Curves

⏱ 2mℹ️ Source: PYQ_2018

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