If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y$ is switched, the result is

A) $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$
B) $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$
C) $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$
D) $\int _ { 0 } ^ { 2 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$
E) $\int _ { 0 } ^ { 1 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$

Question

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y$ is switched, the result is

A) $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$
B) $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$
C) $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$
D) $\int _ { 0 } ^ { 2 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$
E) $\int _ { 0 } ^ { 1 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$

Quizplus · Accepted Answer

Switching the order of integration requires determining the new limits for $x$ and $y$. The original inner integral has $x$ limits from $y^2$ to $2y$, and $y$ ranges from 0 to 2. To switch the order, we find the range of $x$ covered by $y^2$ to $2y$, which is from 0 (when $y=0$) to 4 (when $y=2$). For a given $x$, $y$ ranges from $x/2$ (solving $2y=x$) to $\sqrt{x}$ (solving $y^2=x$). Therefore, the correct switched order of integration is from 0 to 4 in $x$, and for each $x$, from $x/2$ to $\sqrt{x}$ in $y$.

If the Order of Integration Of ∫02[∫y22yf(x,y)dx]dy\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y∫02​[∫y22y​f(x,y)dx]dy

If the Order of Integration Of $\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y$