The area (in sq. units) of the region x , y : x ≥ 0 , x + y ≤ 3 , x 2 ≤ 4 y a n d y ≤ 1 + x is

Required Area = ∫ 0 1 1 + x d x + ∫ 1 2 3 - x d x - ∫ 0 2 x 2 4 d x = x + 2 3 x 3 2 0 1 + 3 x - x 2 2 1 2 - x 3 12 0 2 = 1 + 2 3 + 4 - 5 2 - 2 3 = 5 2 sq . units

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Mathematics - Area Under Curves Question with Solution | TestHub

MathematicsArea Under CurvesArea bounded by Miscellaneous CurvesMedium2 minPYQ_2017

MathematicsMediumsingle choice

The area (in sq. units) of the region $\{(x, y) : x \geq 0, x + y \leq 3, x^{2} \leq 4 y a n d y \leq 1 + \sqrt{x}\}$ is

Options:

Answer:

Solution:

Required Area

$= \int_{0}^{1} (1 + \sqrt{x}) d x + \int_{1}^{2} (3 - x) d x - \int_{0}^{2} \frac{x^{2}}{4} d x$

$= {[x + \frac{2}{3} x^{\frac{3}{2}}]}_{0}^{1} + {[3 x - \frac{x^{2}}{2}]}_{1}^{2} - {[\frac{x^{3}}{12}]}_{0}^{2}$

$= (1 + \frac{2}{3}) + (4 - \frac{5}{2}) - \frac{2}{3}$

$= \frac{5}{2} sq . units$

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

⏱ 2mℹ️ Source: PYQ_2017

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