Evaluate the double integral over the given region.
-$$\iint _ { R } 8 x \sin x y d A , \quad R : 0 \leq x \leq \pi , 0 \leq y \leq 1$$
A) $8 \pi$
B) $\pi$
C) $8 \pi - 8$
D) $\frac { \pi } { 8 }$

Question

Quizplus · Accepted Answer

The integral can be separated into two parts due to its limits and the nature of the function: $\int_{0}^{\pi} 8x dx \int_{0}^{1} \sin(xy) dy$. The first integral evaluates to $4\pi^2$, and the second integral, after substitution and evaluation, results in $\frac{2}{x}$ (from $-\cos(xy)/x$ evaluated from 0 to 1). Multiplying these results gives $8\pi$.

Evaluate the Double Integral Over the Given Region ∬R8xsin⁡xydA,R:0≤x≤π,0≤y≤1\iint _ { R } 8 x \sin x y d A , \quad R : 0 \leq x \leq \pi , 0 \leq y \leq 1∬R​8xsinxydA,R:0≤x≤π,0≤y≤1

Evaluate the Double Integral Over the Given Region $\iint _ { R } 8 x \sin x y d A , \quad R : 0 \leq x \leq \pi , 0 \leq y \leq 1$