Question

The area of the region bounded by $x^2=4y,y=1,y=4$ and the y-axis lying in the first quadrant is ______ square units

• A. $\frac{22}{3}$
• B. $\frac{28}{3}$
• C. $30$
• D. $\frac{21}{4}$

Answer 1

Answer: (B)

$\frac{28}{3}$

The given equations are $x^2=4y$ .......(1)

$y=1$ ......(2)

And $y=4$ ......(3)

Substitute the values of equation (2) and (3) in equation (1)

$x=\mp 2$

Thus, the intersecting co-ordinates are $(-2,1)$ and $(2,1)$ with parabola and line.

$x=\mp 4$

Thus. the intersecting co-ordinates are $(-4,1)$ and $(4,1)$ with parabola and line.

Thus, area of the region bounded by $x^2=4y$, $y=1$, $y=4$ and $y-axis$ lying in the first quadrant is,

$A={\int }_1^{4}x\partial y$

$={\int }_1^{4}2\sqrt{y}\partial y$ [$\because x^2=4y$]

$=2[\frac{y^{(}\frac{3}{2})}{\frac{3}{2}}{]}_1^4$

$=\frac{4}{3}[y^{\frac{3}{2}}{]}_1^4$

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

$=\frac{4}{3}[8-1]$

$=\frac{4}{3}\times 7$

$=\frac{28}{3}$

Thus, $\frac{28}{3}$ unit is the area of the bounded region.