Evaluate the double integral over the given region.
-$$\iint _ { R } ( \sqrt { x } + \sqrt { y } ) d A , \quad R : 0 \leq x \leq 1,0 \leq y \leq 1$$
A) $\frac { 2 } { 3 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 1 } { 3 }$

Question

Quizplus · Accepted Answer

The given double integral can be separated into two integrals due to the linearity of integration. Thus, we have $\int_{0}^{1} \int_{0}^{1} (\sqrt{x} + \sqrt{y}) \,dx\,dy = \int_{0}^{1} \int_{0}^{1} \sqrt{x} \,dx\,dy + \int_{0}^{1} \int_{0}^{1} \sqrt{y} \,dx\,dy$. Evaluating each integral separately, we get $\frac{2}{3}x^{3/2}$ evaluated from 0 to 1 for the x integral and $\frac{2}{3}y^{3/2}$ evaluated from 0 to 1 for the y integral. Each integral evaluates to $\frac{2}{3}$, and since we have two of them, the total is $\frac{2}{3} + \frac{2}{3} = \frac{4}{3}$.

Evaluate the Double Integral Over the Given Region ∬R(x+y)dA,R:0≤x≤1,0≤y≤1\iint _ { R } ( \sqrt { x } + \sqrt { y } ) d A , \quad R : 0 \leq x \leq 1,0 \leq y \leq 1∬R​(x​+y​)dA,R:0≤x≤1,0≤y≤1

Evaluate the Double Integral Over the Given Region $\iint _ { R } ( \sqrt { x } + \sqrt { y } ) d A , \quad R : 0 \leq x \leq 1,0 \leq y \leq 1$