Evaluate the double integral $\iint _ { R } ( x + \sin y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq \pi \}$ .
A) $2 + \pi$

B) $4 + \pi$
C) $( 2 + \pi ) / 2$
D) $4 + 2 \pi$
E) $- 2 - \pi$
F) $- 4 - \pi$

G) $- ( 2 + \pi ) / 2$

H) $- ( 4 + \pi ) / 2$

Question

Evaluate the double integral $\iint _ { R } ( x + \sin y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq \pi \}$ .
A) $2 + \pi$

B) $4 + \pi$
C) $( 2 + \pi ) / 2$ 
D) $4 + 2 \pi$ 
E) $- 2 - \pi$ 
F) $- 4 - \pi$

G) $- ( 2 + \pi ) / 2$

H) $- ( 4 + \pi ) / 2$

Quizplus · Accepted Answer

The double integral can be separated into two integrals: $\int_{0}^{2} x dx$ and $\int_{0}^{\pi} \sin y dy$. The first integral evaluates to $x^2/2$ from 0 to 2, which is $2^2/2 - 0 = 2$. The second integral evaluates to $-\cos y$ from 0 to $\pi$, which is $-\cos(\pi) - (-\cos(0)) = 2$. Therefore, the total integral is $2 \times 2 + 2 = 4 + 2\pi$.

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Evaluate the Double Integral $\iint _ { R } ( x + \sin y ) d A$