Integrate the function f over the given region.
-$f ( x , y ) = 4 x \sin x y$ over the rectangle $0 \leq x \leq \pi , 0 \leq y \leq 1$
A) $4 \pi$
B) $\pi$
C) $\frac { \pi } { 4 }$
D) $4 \pi - 4$

Question

Quizplus · Accepted Answer

The integral of $f(x, y) = 4x \sin(xy)$ over the rectangle $0 \leq x \leq \pi, 0 \leq y \leq 1$ can be computed by first integrating with respect to $y$ and then with respect to $x$. The integration process involves using the fact that the integral of $x \sin(xy)$ with respect to $y$ gives a function that, when evaluated from $0$ to $1$, results in a form that simplifies nicely upon integration over $x$. Specifically, the integral with respect to $y$ yields $-\frac{4}{x}\cos(xy)$ evaluated from $0$ to $1$, which simplifies to $-\frac{4}{x}(\cos(x) - 1)$. Integrating this expression with respect to $x$ from $0$ to $\pi$ and applying the fundamental theorem of calculus, we find that the result is $4\pi$, as the negative cosine term integrates to $0$ over this interval, and the remaining terms simplify to the given answer.

Integrate the Function F Over the Given Region f(x,y)=4xsin⁡xyf ( x , y ) = 4 x \sin x yf(x,y)=4xsinxy

Integrate the Function F Over the Given Region $f ( x , y ) = 4 x \sin x y$