Use the given transformation to evaluate the integral.
-$$\begin{array} { l }
x = 10 u , y = 7 v , z = 2 w ; \
\quad \int _ { R } \int _ { \int } \int ^ { 2 } y ^ { 2 } d x d y d z
\end{array}$$
where $R$ is the interior of the ellipsoid $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 49 } + \frac { z ^ { 2 } } { 4 } = 1$
A) $\frac { 16 } { 3 } \pi$
B) $7 \pi$
C) $\frac { 28 } { 5 } \pi$
D) $\frac { 28 } { 3 } \pi$

Question

Use the given transformation to evaluate the integral.
-$$\begin{array} { l } 
x = 10 u , y = 7 v , z = 2 w ; \
\quad \int _ { R } \int _ { \int } \int ^ { 2 } y ^ { 2 } d x d y d z
\end{array}$$
where $R$ is the interior of the ellipsoid $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 49 } + \frac { z ^ { 2 } } { 4 } = 1$
A) $\frac { 16 } { 3 } \pi$
B) $7 \pi$
C) $\frac { 28 } { 5 } \pi$
D) $\frac { 28 } { 3 } \pi$

Quizplus · Accepted Answer

The Jacobian of the transformation is $J = | \frac { \partial ( x , y , z ) } { \partial ( u , v , w ) } | = | 10 \quad 0 \quad 0 | = 10$So, the integral becomes$$\begin{array} { l } \int _ { R } \int _ { \int } \int ^ { 2 } y ^ { 2 } d x d y d z \\= \int _ { R } \int _ { \int } \int ^ { 2 } ( 7 v ) ^ { 2 } 10 d v d u d w \\= 7 ^ { 2 } \cdot 10 \int _ { R } \int _ { \int } \int ^ { 2 } v ^ { 2 } d v d u d w \\= 490 \int _ { R } \int _ { \int } \left[ \frac { v ^ { 3 } } { 3 } \right] _ { 0 } ^ { 2 } d u d w \\= 490 \int _ { R } \int _ { \int } \frac { 8 } { 3 } d u d w \\= \frac { 490 \cdot 8 } { 3 } \int _ { R } d u d w \\= \frac { 16 } { 3 } \int _ { R } d A \\= \frac { 16 } { 3 } \pi \left( \frac { 10 } { 2 } \right) \left( \frac { 7 } { 2 } \right) \\= \frac { 16 } { 3 } \pi \cdot 5 \cdot 7 \\= \frac { 16 } { 3 } \pi \cdot 35 \\= \frac { 16 } { 3 } \pi\end{array}$$

Use the Given Transformation to Evaluate the Integral x=10u,y=7v,z=2w;∫R∫∫∫2y2dxdydz\begin{array} { l } x = 10 u , y = 7 v , z = 2 w ; \\\quad \int _ { R } \int _ { \int } \int ^ { 2 } y ^ { 2 } d x d y d z\end{array}x=10u,y=7v,z=2w;∫R​∫∫​∫2y2dxdydz​

Use the Given Transformation to Evaluate the Integral $\begin{array} { l } x = 10 u , y = 7 v , z = 2 w ; \\\quad \int _ { R } \int _ { \int } \int ^ { 2 } y ^ { 2 } d x d y d z\end{array}$