Let $f ( x , y ) = x y$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into four subrectangles by the lines $x = \frac { 1 } { 2 }$ and $y = \frac { 1 } { 2 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the upper right corner 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 $f ( x , y ) = x y$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into four subrectangles by the lines $x = \frac { 1 } { 2 }$ and $y = \frac { 1 } { 2 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the upper right corner 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 partition divides the unit square into four equal subrectangles, each with dimensions $\frac{1}{2} \times \frac{1}{2}$. The upper right corners of these subrectangles are $(\frac{1}{2}, \frac{1}{2})$, $(1, \frac{1}{2})$, $(\frac{1}{2}, 1)$, and $(1, 1)$. Evaluating $f(x, y) = xy$ at these points gives $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{2}$, and $1$, respectively. The sum of these values is $\frac{9}{4}$ times the area of each subrectangle, which is $\frac{1}{4}$, leading to a total Riemann sum of $\frac{9}{16}$.

Let f(x,y)=xyf ( x , y ) = x yf(x,y)=xy , and Let

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