Question

By reversing the order of integration. ${\int }_0^2{\int }_{x^2}^{2x}f(x,y)dydx$ may be resperesented as

• A. ${\int }_0^2{\int }_{2x}^{x^2}f(x,y)dydx$
• B. ${\int }_0^2{\int }_y^{\sqrt{y}}(x,y)dxdy$
• C. ${\int }_0^4{\int }_{y/2}^{\sqrt{y}}(x,y)dxdy$
• D. ${\int }_{x^2}^{2x}{\int }_0^2(x,y)dydx$

Answer 1

The correct option is C ${\int }_0^4{\int }_{y/2}^{\sqrt{y}}(x,y)dxdy$

$I={\int }_0^2{\int }_{x^2}^{2x}f(x,y)dydx$
To reverse the order of integration we will have to form horizontal strip
$y\to x^2$ to $2x$
$x\to 0$ to $2$



$I={\int }_0^4{\int }_{y/2}^{\sqrt{y}}f(x,y)dxdy$
Hence (c) is the correct option