The area of the region given by x , y : x y ≤ 8 , 1 ≤ y ≤ x 2 is :

Mathematics - Area Under Curves Question with Solution | TestHub

MathematicsArea Under CurvesArea bounded by two curvesMedium2 minPYQ_2023

MathematicsMediumsingle choice

The area of the region given by $\{(x, y) : x y \leq 8, 1 \leq y \leq x^{2}\}$ is :

Answer:

Solution:

Given,

$x y \leq 8 & 1 \leq y \leq x^{2}$

On solving above curve we get intersecting point as $(2, 4)$

Now on plotting the diagram of the given curves we get,

Required area

$A = \int_{1}^{2} (x^{2} - 1) d x + \int_{2}^{8} (\frac{8}{x} - 1) d x$

$\Rightarrow A = \int_{1}^{2} x^{2} d x + \int_{2}^{8} \frac{8}{x} d x - \int_{1}^{8} d x$

$\Rightarrow A = {[\frac{x^{3}}{3}]}_{1}^{2} + 8 {[\ln x]}_{2}^{8} - {[x]}_{1}^{8}$

$\Rightarrow A = 8 \ln 4 - \frac{14}{3}$

$\Rightarrow A = 16 \ln 2 - \frac{14}{3}$ sq. units

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

⏱ 2mℹ️ Source: PYQ_2023

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