If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x$ is switched, the result is

A) $\int _ { 0 } ^ { 8 } \left[ \int _ { \sqrt [ 4 ] { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$
B) $\int _ { 0 } ^ { 8 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 2 ] { y } } f ( x , y ) d x \right] d y$
C) $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$
D) $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$
E) $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$

Question

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x$ is switched, the result is

A) $\int _ { 0 } ^ { 8 } \left[ \int _ { \sqrt [ 4 ] { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$
B) $\int _ { 0 } ^ { 8 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 2 ] { y } } f ( x , y ) d x \right] d y$
C) $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$
D) $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$
E) $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$

Quizplus · Accepted Answer

To switch the order of integration, we first identify the bounds for $y$ and $x$. From the original integral, $y$ ranges from $x^2$ to $4x$, and $x$ ranges from 0 to 2. Solving $x^2 = 4x$ gives $x = 0$ and $x = 4$, but since $x$ is limited to 2, the upper bound for $y$ is $4(2) = 8$. Thus, $y$ ranges from 0 to 8. For a given $y$, $x$ ranges from $y/4$ (from $y = 4x$) to $\sqrt{y}$ (from $y = x^2$). Therefore, the correct switched order of integration is $\int _ { 0 } ^ { 8 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 2 ] { y } } f ( x , y ) d x \right] d y$.

If the Order of Integration Of ∫02[∫x24xf(x,y)dy]dx\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x∫02​[∫x24x​f(x,y)dy]dx

If the Order of Integration Of $\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x$