Area bounded by the parabola y = 4x2, Y-axis and the lines y = 1, y = 4 is:

Question

Area bounded by the parabola $y = 4 x^{2},$ Y-axis and the lines $y = 1, y = 4$ is:

Smart Achievers · Accepted Answer

(3)

Ans. $\frac{7}{3}$ sq. units

Explanation :

Given, $y = 4 x^{2}$

Area to be found between Y-axis, $y = 1$ and $y = 4$ .

The figure is as follows:

At $y = 1$ $\to y = 4 x^{2}$ gives $x = \frac{1}{2}$ and $- \frac{1}{2}$

At $y = 4$ $\to y = 4 x^{2}$ gives $4 = 4 x^{2}$ or $x = 1, - 1$

Here,

$Area = \int_{1}^{4} x d y = \int_{1}^{4} \frac{\sqrt{y}}{2} d y$

$= \frac{1}{2} \int_{1}^{4} y^{1 / 2} d y$

$= \frac{1}{2} {[\frac{2}{3} y^{3 / 2}]}_{1}^{4}$ $= \frac{1}{3} {[y^{3 / 2}]}_{1}^{4}$

$= \frac{1}{3} [4^{3 / 2} - 1^{3 / 2}]$

$= \frac{1}{3} (8 - 1) = \frac{7}{3} sq. units.$

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