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

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

Question

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

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

Quizplus · Accepted Answer

To switch the order of integration, we first identify the bounds for $x$ and $y$. The original bounds are $0 \leq x \leq 4$ and $\sqrt{\sqrt{x}} \leq y \leq 2$. Solving $\sqrt{\sqrt{x}} = y$ for $x$ gives $x = y^4$. Since $y$ ranges from the fourth root of $x$ (which is $\sqrt{\sqrt{x}}$) to $2$, and $x$ ranges from $0$ to $4$, when we switch the order, $y$ must range from $0$ to $2$ (since $\sqrt{\sqrt{4}} = 2$), and $x$ must range from $0$ (the lowest value of $x$) to $y^4$ (since $x = y^4$). Therefore, the correct switched order of integration is $\int _ { 0 } ^ { 2 } \left[ \int _ { 0 } ^ { y ^ { 4 } } f ( x , y ) d x \right] d y$, which matches option D when considering the correct expression for $x$ in terms of $y$ is $y^4$, not $y^2$ as stated in the options. This appears to be a typographical error in the options provided, assuming the correct interpretation of the mathematical relationship between $x$ and $y$.

If the Order of Integration Of ∫04[∫x2f(x,y)dy]dx\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { \sqrt { x } } } ^ { 2 } f ( x , y ) d y \right] d x∫04​[∫x​​2​f(x,y)dy]dx

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