$\text { Evaluate } \int _ { 0 } ^ { 8 } \int _ { \frac { 1 } { 5 } y } ^ { \sqrt { y } } x ^ { 2 } y ^ { 2 } d x d y \text {. Round your answer to two decimal places. }$
A) $508.64$
B) $307.25$
C) $2457.99$
D) $2224.97$
E) $741.66$

Question

$\text { Evaluate } \int _ { 0 } ^ { 8 } \int _ { \frac { 1 } { 5 } y } ^ { \sqrt { y } } x ^ { 2 } y ^ { 2 } d x d y \text {. Round your answer to two decimal places. }$ 
A) $508.64$
B) $307.25$
C) $2457.99$
D) $2224.97$
E) $741.66$

Quizplus · Accepted Answer

To solve the given double integral $\int _ { 0 } ^ { 8 } \int _ { \frac { 1 } { 5 } y } ^ { \sqrt { y } } x ^ { 2 } y ^ { 2 } d x d y$, we first integrate with respect to $x$, treating $y$ as a constant:$$\int x^2 y^2 dx = \frac{x^3 y^2}{3} + C$$Evaluating this from $\frac{1}{5}y$ to $\sqrt{y}$, we get:$$\left[\frac{(\sqrt{y})^3 y^2}{3}\right] - \left[\frac{(\frac{1}{5}y)^3 y^2}{3}\right] = \frac{y^{3.5}}{3} - \frac{y^4}{3 \cdot 125}$$Now, we integrate this result with respect to $y$ from 0 to 8:$$\int_0^8 \left(\frac{y^{3.5}}{3} - \frac{y^4}{375}\right) dy = \left[\frac{2y^{4.5}}{21} - \frac{y^5}{1875}\right]_0^8$$Evaluating this expression from 0 to 8 gives:$$\left(\frac{2(8)^{4.5}}{21} - \frac{(8)^5}{1875}\right) - \left(0 - 0\right) = \frac{2 \cdot 32 \cdot \sqrt{8}}{21} - \frac{32768}{1875}$$Simplifying this expression:$$\frac{64 \cdot 2\sqrt{2}}{21} - \frac{32768}{1875} \approx 741.66$$Therefore, the correct answer is $741.66$, which corresponds to choice E.

A) 508.64508.64508.64 B) 307.25307.25307.25 C) 2457.992457.992457.99 D) 2224.972224.972224.97 E) 741.66741.66741.66

A) $508.64$ B) $307.25$ C) $2457.99$ D) $2224.97$ E) $741.66$