Find the area of the surface given by $\mathbf { r } ( u , v ) = 9 u \mathbf { i } - v \mathbf { j } + v \mathbf { k } \text {, where }$ $0 \leq u \leq 2$ and $0 \leq v \leq 4$
A) $81 \sqrt { 2 }$
B) $38 \sqrt { 2 }$
C) $72 \sqrt { 2 }$
D) $45 \sqrt { 2 }$
E) $22 \sqrt { 2 }$

Question

Quizplus · Accepted Answer

The area of the surface given by a parametric surface $\mathbf{r}(u, v)$ can be found using the formula $\int \int ||\mathbf{r}_u \times \mathbf{r}_v|| dudv$, where $\mathbf{r}_u$ and $\mathbf{r}_v$ are the partial derivatives of $\mathbf{r}$ with respect to $u$ and $v$, respectively. For $\mathbf{r}(u, v) = 9u\mathbf{i} - v\mathbf{j} + v\mathbf{k}$, we find $\mathbf{r}_u = 9\mathbf{i}$ and $\mathbf{r}_v = -\mathbf{j} + \mathbf{k}$. The cross product $\mathbf{r}_u \times \mathbf{r}_v = 9\mathbf{j} + 9\mathbf{k}$, and its magnitude is $||\mathbf{r}_u \times \mathbf{r}_v|| = \sqrt{81 + 81} = 9\sqrt{2}$. Integrating over the given bounds for $u$ and $v$, the area is $\int_0^2 \int_0^4 9\sqrt{2} dvdu = 9\sqrt{2} \times 8 = 72\sqrt{2}$.

Find the Area of the Surface Given By r(u,v)=9ui−vj+vk, where \mathbf { r } ( u , v ) = 9 u \mathbf { i } - v \mathbf { j } + v \mathbf { k } \text {, where }r(u,v)=9ui−vj+vk, where

Find the Area of the Surface Given By $\mathbf { r } ( u , v ) = 9 u \mathbf { i } - v \mathbf { j } + v \mathbf { k } \text {, where }$