Let $f ( x , y ) = x ^ { 2 }$ , 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 $x = \frac { 1 } { 2 }$ , 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 $f ( x , y ) = x ^ { 2 }$ , 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 $x = \frac { 1 } { 2 }$ , 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 partition divides R into two subrectangles: $R_1$ with $0 \leq x \leq \frac{1}{2}$ and $R_2$ with $\frac{1}{2} \leq x \leq 1$, both with $0 \leq y \leq 1$. The centers of these subrectangles are $\left(\frac{1}{4}, \frac{1}{2}\right)$ and $\left(\frac{3}{4}, \frac{1}{2}\right)$ respectively. The double Riemann sum is calculated by evaluating $f$ at these centers and multiplying by the area of each subrectangle ($\frac{1}{2} \times 1 = \frac{1}{2}$ for each). Thus, the sum is $\left(\left(\frac{1}{4}\right)^2 + \left(\frac{3}{4}\right)^2\right) \times \frac{1}{2} = \left(\frac{1}{16} + \frac{9}{16}\right) \times \frac{1}{2} = \frac{5}{16}$.

Let f(x,y)=x2f ( x , y ) = x ^ { 2 }f(x,y)=x2

Let $f ( x , y ) = x ^ { 2 }$