Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { - 1 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } f ( x , y ) d y d x$
A) $\int _ { 0 } ^ { 1 } \int _ { - \sqrt { y } } ^ { \sqrt { y } } f ( x , y ) d x d y$
B) $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { y } } f ( x , y ) d x d y$
C) $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { y } } f ( x , y ) d x d y$
D) $\int _ { - 1 } ^ { 1 } \int _ { - y ^ { 2 } } ^ { y ^ { 2 } } f ( x , y ) d x d y$

Question

Quizplus · Accepted Answer

The original integral limits for $y$ go from $x^2$ to 1, meaning $y$ ranges from 0 (when $x=0$) to 1 (the upper limit for $y$). For a given $y$, $x$ ranges from $-\sqrt{y}$ to $\sqrt{y}$, reflecting the bounds where $x^2 = y$. This is correctly represented in option A, where the outer integral is over $y$ from 0 to 1, and the inner integral is over $x$ from $-\sqrt{y}$ to $\sqrt{y}$.

Give the Iterated Integral as an Iterated Integral or Sum ∫−11∫x21f(x,y)dydx\int _ { - 1 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } f ( x , y ) d y d x∫−11​∫x21​f(x,y)dydx

Give the Iterated Integral as an Iterated Integral or Sum $\int _ { - 1 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } f ( x , y ) d y d x$