The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, the value of a is:

Question

The area between $x = y^{2}$ and $x = 4$ is divided into two equal parts by the line $x = a$ , the value of $a$ is:

Smart Achievers · Accepted Answer

(2)

Ans. $\sqrt[3]{16}$

Explanation :

Given, $x = y^{2}$ is a parabola symmetric to the positive $X$ -axis with vertex (0, 0).

We have given that,

$\int_{0}^{a} \sqrt{x} d x = \int_{a}^{4} \sqrt{x} d x$

$\Rightarrow$ ${[\frac{x^{3 / 2}}{\frac{3}{2}}]}_{0}^{a} = {[\frac{x^{3 / 2}}{\frac{3}{2}}]}_{a}^{4}$

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

$\Rightarrow$ $\frac{2}{3} a^{3 / 2} + \frac{2}{3} a^{3 / 2} = \frac{2}{3} \times 8$

$\Rightarrow$ $\frac{4}{3} a^{3 / 2} = \frac{16}{3}$

$\Rightarrow$ $a \sqrt{a} = \frac{16}{3} \times \frac{3}{4}$

$\Rightarrow$ $a^{3 / 2} = 4$

$\Rightarrow$ $a = 4^{2 / 3} = {(2^{2})}^{2 / 3}$

$\Rightarrow$ $a = 2^{4 / 3} = \sqrt[3]{16}$

Want Us to Email you About Special Offers & Updates?