Solve the problem.
-Integrate $f ( x , y ) = \sin \left( x ^ { 2 } + y ^ { 2 } \right)$ over the region $0 \leq x ^ { 2 } + y ^ { 2 } \leq 4$.
A) $2 \pi \left( 1 - \cos ^ { 2 } 2 \right)$
B) $\pi \left( 1 - \cos ^ { 2 } 2 \right)$
C) $2 \pi ( 1 - \cos 4 )$
D) $\pi ( 1 - \cos 4 )$

Question

Quizplus · Accepted Answer

The integral of $f(x, y) = \sin(x^2 + y^2)$ over the region defined by $0 \leq x^2 + y^2 \leq 4$ can be solved using polar coordinates, where $x^2 + y^2 = r^2$ and the limits for $r$ are from 0 to 2 (since $x^2 + y^2 \leq 4$ implies $r^2 \leq 4$, or $r \leq 2$). The integral becomes $\int_0^{2\pi} \int_0^2 r \sin(r^2) dr d\theta$. Solving this gives $\pi(1 - \cos(4))$, which corresponds to choice D.

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