Let $f ( x , y ) = x$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into two subrectangles by the line $x = \frac { 1 } { 4 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the center of Rij. Calculate the double Riemann sum of f.
A) $\frac { 3 } { 16 }$

B) $\frac { 1 } { 4 }$
C) $\frac { 5 } { 16 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 7 } { 16 }$
F) $\frac { 1 } { 2 }$

G) $\frac { 9 } { 16 }$

H) $\frac { 3 } { 4 }$

Question

Let $f ( x , y ) = x$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into two subrectangles by the line $x = \frac { 1 } { 4 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the center of Rij. Calculate the double Riemann sum of f.
A) $\frac { 3 } { 16 }$

B) $\frac { 1 } { 4 }$
C) $\frac { 5 } { 16 }$ 
D) $\frac { 3 } { 8 }$
E) $\frac { 7 } { 16 }$  
F) $\frac { 1 } { 2 }$ 
 
G) $\frac { 9 } { 16 }$

H) $\frac { 3 } { 4 }$

Quizplus · Accepted Answer

The region R is divided into two subrectangles: one from $x = 0$ to $x = \frac{1}{4}$ and the other from $x = \frac{1}{4}$ to $x = 1$. The centers of these subrectangles are $(\frac{1}{8}, \frac{1}{2})$ and $(\frac{5}{8}, \frac{1}{2})$. The Riemann sum is $\frac{1}{4} \times 1 \times \frac{1}{8} + \frac{3}{4} \times 1 \times \frac{5}{8} = \frac{1}{32} + \frac{15}{32} = \frac{16}{32} = \frac{1}{2}$.

Let f(x,y)=xf ( x , y ) = xf(x,y)=x , and Let

Let $f ( x , y ) = x$ , and Let