Mathematics - Area Under Curves Question with Solution | TestHub

MathematicsArea Under CurvesArea bounded by two curvesMedium2 minPYQ_2020

MathematicsMediumsingle choice

Consider a region $R = \{(x, y) \in R^{2} : x^{2} \leq y \leq 2 x\}$ . If a line $y = α$ divides the area of region $R$ into two equal parts, then which of the following is true ?

Options:

Answer:

Solution:

$\int_{0}^{4} (\sqrt{y} - \frac{y}{2}) d y = 2 \int_{0}^{α} (\sqrt{y} - \frac{y}{2}) d y$

${(\frac{2 y^{3 / 2}}{3} - \frac{y^{2}}{2})}_{0}^{4} = {2 (\frac{2 y^{3 / 2}}{3} - \frac{y^{2}}{4})}_{0}^{α}$

$\Rightarrow \frac{16}{3} - 4 = 2 (\frac{2}{3} α^{3 / 2} - \frac{α^{2}}{4})$

$\Rightarrow \frac{4}{3} = 2 (\frac{2}{3} α^{3 / 2} - \frac{α^{2}}{4})$

$\Rightarrow \frac{2}{3} = \frac{8 α^{3 / 2} - 3 α^{2}}{12}$

$\Rightarrow 8 α^{3 / 2} - 3 α^{2} = 8$

$\Rightarrow 3 α^{2} - 8 α^{3 / 2} + 8 = 0$

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

⏱ 2mℹ️ Source: PYQ_2020

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