Evaluate the surface integral $$S$(2$yj$+$zk$)$dS$$S$(2$yj$+$zk$)$dS$.$

Where SS consists of the paraboloid y$=$x$2+$z$2,0$y$1$y$=$x$2+$z$2,0$y$1$ and the disk x$2+$z$21,$y$=1$x$2+$z$21,$y$=1,$ and has outward orientation.

Consider splitting the surface into its partsS$=$S$1$S$2$S$=$S$1$S$2$whereS$1$S$1$is the paraboloid andS$2$S$2$is the disk. Then we have

$$SF$$dS$=$S$1$F$$dS$1+$S$2$F$$dS$2$SF$$dS$=$S$1$F$$dS$1+$S$2$F$$dS$2$

S$1$S$1$can be parametrized byr$1($s$,$t$)=$scos$($t$),$r$1($s$,$t$)=$scos$($t$),$

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$,$

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$$

S$2$S$2$can be parametrized byr$2($s$,$t$)=$scos$($t$),$r$2($s$,$t$)=$scos$($t$),$

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$,$

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$$

$$S$(2$yj$+$zk$)$dS$=$t$2$t$1$s$2$s$1$S$(2$yj$+$zk$)$dS$=$t$1$t$2$s$1$s$2$

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ds$$dt$+$t$2$t$1$s$2$s$1$ds$$dt$+$t$1$t$2$s$1$s$2$

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ds$$dtds$$dt

$($EvaluateS$1$S$1$for the first blank andS$2$S$2$for the second blank$)$

where

t$1=$t$1=$

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s$1=$s$1=$

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t$2=$t$2=$

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s$2=$s$2=$

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Evaluate

$$S$(2$yj$+$zk$)$dS$=$S$(2$yj$+$zk$)$dS$=$

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$++$

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$($EvaluateS$1$S$1$for the first blank andS$2$S$2$for the second blank$)$