Evaluate the integral $\iint _ { R } \sqrt { x ^ { 2 } + y ^ { 2 } } d A$ , where R is the disk with center the origin and radius 2.
A) $\frac { \pi } { 3 }$

B) $\frac { 2 \pi } { 3 }$
C) $\frac { 4 \pi } { 3 }$
D) $\frac { 8 \pi } { 3 }$
E) $\frac { 16 \pi } { 3 }$
F) $\frac { 32 \pi } { 3 }$

G) $\frac { 64 \pi } { 3 }$

H) $\frac { 128 \pi } { 3 }$

Question

Evaluate the integral $\iint _ { R } \sqrt { x ^ { 2 } + y ^ { 2 } } d A$ , where R is the disk with center the origin and radius 2.
A) $\frac { \pi } { 3 }$

B) $\frac { 2 \pi } { 3 }$ 
C) $\frac { 4 \pi } { 3 }$ 
D) $\frac { 8 \pi } { 3 }$
E) $\frac { 16 \pi } { 3 }$ 
F) $\frac { 32 \pi } { 3 }$

G) $\frac { 64 \pi } { 3 }$

H) $\frac { 128 \pi } { 3 }$

Quizplus · Accepted Answer

The integral can be evaluated using polar coordinates, where $ x = r \cos \theta $ and $ y = r \sin \theta $. The region R is a disk of radius 2, so $ 0 \leq r \leq 2 $ and $ 0 \leq \theta \leq 2\pi $. The integrand becomes $ r $ and the area element $ dA = r \, dr \, d\theta $. The integral is $\int_0^{2\pi} \int_0^2 r^2 \, dr \, d\theta = \frac{16\pi}{3}$.

Evaluate the Integral ∬Rx2+y2dA\iint _ { R } \sqrt { x ^ { 2 } + y ^ { 2 } } d A∬R​x2+y2​dA

Evaluate the Integral $\iint _ { R } \sqrt { x ^ { 2 } + y ^ { 2 } } d A$