The area of the region x , y : x 2 ≤ y ≤ x 2 - 4 , y ≥ 1 is

The required diagram is: The shaded region is symmetric about y - axis. Required area = 2 ∫ 1 2 y d y + ∫ 2 4 4 - y d y = 2 y 3 / 2 3 2 1 2 - 2 ( 4 - y ) 3 / 2 3 2 4 = 4 3 ( 4 2 - 1 ) Hence this is the correct option.

Mathematics - Area Under Curves Question with Solution | TestHub

MathematicsArea Under CurvesArea bounded by two curvesHard2 minPYQ_2023

MathematicsHardsingle choice

The area of the region $x, y : x^{2} \leq y \leq |x^{2} - 4|, y \geq 1$ is

Options:

Answer:

Solution:

The required diagram is:
The shaded region is symmetric about $y -$ axis.
Required area $= 2 [\int_{1}^{2} \sqrt{y} d y + \int_{2}^{4} \sqrt{4 - y} d y]$
$= 2 {[\frac{y^{3 / 2}}{\frac{3}{2}}]}_{1}^{2} - {[\frac{2 (4 - y)^{3 / 2}}{3}]}_{2}^{4}$
$= \frac{4}{3} (4 \sqrt{2} - 1)$
Hence this is the correct option.

Stream:JEESubject:MathematicsTopic:Area Under CurvesSubtopic:Area bounded by two curves

⏱ 2mℹ️ Source: PYQ_2023

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