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Consider the surface integral $\int_{(S)} xyz \, dS$ where $S$ is the cone with parametric equations $\newline$ $x=u\cos v$ , $\newline$ $y=u\sin v$ , $\newline$ $z=u$ $\newline$ for $0 \leq u \leq 2$ and $0 \leq v \leq \frac{\pi}{2}$ .

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Find derivatives of sine and cosine functions

Full solution

Q. Consider the surface integral

\int_{(S)} xyz \, dS

where

S

is the cone with parametric equations

\newline

x=u\cos v

,

\newline

y=u\sin v

,

\newline

z=u

\newline

for

0 \leq u \leq 2

and

0 \leq v \leq \frac{\pi}{2}

.

Identify Parametric Equations: Identify the parametric equations and the limits for $u$ and $v$ . The cone is given by $x = u \cos(v)$ , $y = u \sin(v)$ , and $z = u$ . The limits are $0 \leq u \leq 2$ and $0 \leq v \leq \frac{\pi}{2}$ .
Express $xyz$ in Terms: Express $xyz$ in terms of $u$ and $v$ using the parametric equations. $\newline$ $xyz = (u \cos(v))(u \sin(v))(u) = u^3 \cos(v) \sin(v)$ .
Calculate Differential Surface Element: Calculate the differential surface element $dS$ for the cone. $\newline$ The cross product of the partial derivatives of the position vector $\mathbf{r}(u, v) = (u \cos(v), u \sin(v), u)$ with respect to $u$ and $v$ gives the magnitude of $dS$ . $\newline$ $\frac{d\mathbf{r}}{du} = (\cos(v), \sin(v), 1)$ , $\frac{d\mathbf{r}}{dv} = (-u \sin(v), u \cos(v), 0)$ . $\newline$ Cross product: $\begin{vmatrix} i & j & k \ \cos(v) & \sin(v) & 1\ -u \sin(v) & u \cos(v) & 0\end{vmatrix} = (u \cos(v) + u \sin(v))i - (u \cos(v))j + (u)k.$ $\newline$ Magnitude of $dS = \sqrt{(u \cos(v) + u \sin(v))^2 + (-u \cos(v))^2 + (u)^2}$ .

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