Question

The double integral ${\int }_0^a{\int }_0^{y}f(x,y)dxdy$ is equivalent to

• A. ${\int }_0^x{\int }_0^{y}f(x,y)dxdy$
• B. ${\int }_0^a{\int }_x^{y}f(x,y)dxdy$
• C. ${\int }_0^a{\int }_x^{a}f(x,y)dydx$
• D. ${\int }_0^a{\int }_x^{a}f(x,y)dydy$

Answer 1

The correct option is C ${\int }_0^a{\int }_x^{a}f(x,y)dydx$



Region of intgeration is bounded by $y=0$ to $y=a$ and $x=0$ to $x=y$.
$\because y$ has constant limits so strip used in given integral is Horizontal.
Hence to change order of integration we will from vertical strip for which, variation of limits will be as follows:
$x=0$ to $x=a\&y=x$ to $y=a$
So $I={\int }_0^a{\int }_0^{y}f(x,y)dxdy={\int }_0^a{\int }_x^{a}f(x,y)dydx$