Solve the problem.
-Write
$$\int _ { 0 } ^ { 8 } \int _ { 0 } ^ { y } f ( x , y ) d x d y + \int _ { 8 } ^ { 16 } \int _ { 0 } ^ { 16 - y } f ( x , y ) d x d y + \int _ { 0 } ^ { 8 } \int _ { y } ^ { 8 } f ( x , y ) d x d y + \int _ { 0 } ^ { 8 } \int _ { 8 } ^ { 16 - y } f ( x , y ) d x d y$$
as a single iterated integral with the same order of integration.
A) $\int _ { 8 } ^ { 16 } \int _ { 8 } ^ { 16 - y } f ( x , y ) d x d y$
B) $\int _ { 8 } ^ { 16 } \int _ { 0 } ^ { 16 - y } f ( x , y ) d x d y$
C) $\int _ { 0 } ^ { 16 } \int _ { 0 } ^ { 16 - y } f ( x , y ) d x d y$
D) $\int _ { 0 } ^ { 16 } \int _ { 8 } ^ { 16 - y } f ( x , y ) d x d y$

Question

Quizplus · Accepted Answer

To combine the given integrals into a single iterated integral, we need to understand the regions of integration they represent in the xy-plane. The first integral covers the area where $0 \leq y \leq 8$ and $0 \leq x \leq y$, the second where $8 \leq y \leq 16$ and $0 \leq x \leq 16 - y$, the third where $0 \leq y \leq 8$ and $y \leq x \leq 8$, and the fourth where $0 \leq y \leq 8$ and $8 \leq x \leq 16 - y$. Combining these regions, we cover the entire square region where $0 \leq y \leq 16$ and $0 \leq x \leq 16 - y$, which corresponds to choice C.

Solve the Problem As a Single Iterated Integral with the Same Order of