Evaluate the double integral $\iint _ { R } \frac { y } { ( x + Y ) ^ { 2 } } d A$ , where $R = \{ ( x , y ) \mid 1 \leq x \leq 2,1 \leq y \leq 2 \}$ .
A) $\frac { 9 } { 8 }$

B) $\frac { 1 } { 2 }$
C) $- \ln \frac { 1 } { 2 }$
D) $\ln \frac { 9 } { 8 }$
E) $\frac { 8 } { 9 }$
F) $\frac { 3 } { 2 }$

G) $\ln \frac { 3 } { 2 }$

H) $\ln \frac { 8 } { 9 }$

Question

Evaluate the double integral $\iint _ { R } \frac { y } { ( x + Y ) ^ { 2 } } d A$ , where $R = \{ ( x , y ) \mid 1 \leq x \leq 2,1 \leq y \leq 2 \}$ .
A) $\frac { 9 } { 8 }$

B) $\frac { 1 } { 2 }$ 
C) $- \ln \frac { 1 } { 2 }$ 
D) $\ln \frac { 9 } { 8 }$ 
E) $\frac { 8 } { 9 }$ 
F) $\frac { 3 } { 2 }$

G) $\ln \frac { 3 } { 2 }$

H) $\ln \frac { 8 } { 9 }$

Quizplus · Accepted Answer

To solve the given double integral $\iint _ { R } \frac { y } { ( x + y ) ^ { 2 } } d A$ over the region $R = \{ ( x , y ) \mid 1 \leq x \leq 2,1 \leq y \leq 2 \}$, we first integrate with respect to $x$ and then with respect to $y$, or vice versa, depending on the simplicity of the integrand after partial integration. The integral becomes:$$\int_{1}^{2} \int_{1}^{2} \frac{y}{(x+y)^2} dx dy$$Let's integrate with respect to $x$ first:$$\int_{1}^{2} \left( \int_{1}^{2} \frac{y}{(x+y)^2} dx \right) dy$$The inner integral with respect to $x$ can be solved by treating $y$ as a constant, which gives us:$$- \frac{y}{x+y} \Big|_{1}^{2} = -y \left( \frac{1}{2+y} - \frac{1}{1+y} \right)$$Simplifying, we get:$$-y \left( \frac{1 - (2+y)}{(1+y)(2+y)} \right) = \frac{y}{(1+y)(2+y)}$$Now, integrate with respect to $y$:$$\int_{1}^{2} \frac{y}{(1+y)(2+y)} dy$$This can be simplified by partial fraction decomposition or recognizing it as a derivative of a logarithmic function. The result of this integration will lead to a logarithmic expression. After performing the integration, the result is:$$\ln \frac{(1+y)}{(2+y)} \Big|_{1}^{2} = \ln \frac{3}{4} - \ln \frac{2}{3} = \ln \frac{3}{4} \times \frac{3}{2} = \ln \frac{9}{8}$$Therefore, the correct answer is $\ln \frac{9}{8}$, which corresponds to choice D.

Evaluate the Double Integral ∬Ry(x+Y)2dA\iint _ { R } \frac { y } { ( x + Y ) ^ { 2 } } d A∬R​(x+Y)2y​dA

Evaluate the Double Integral $\iint _ { R } \frac { y } { ( x + Y ) ^ { 2 } } d A$