009B Sample Final 2, Problem 5

(a) Find the area of the surface obtained by rotating the arc of the curve

$y^{3}=x$

between $(0,0)$ and $(1,1)$ about the $y$-axis.

(b) Find the length of the arc

$y=1+9x^{\frac {3}{2}}$

between the points $(1,10)$ and $(4,73).$

Foundations:
1. The formula for the length  $L$  of a curve  $y=f(x)$  where  $a\leq x\leq b$  is
$L=\int _{a}^{b}{\sqrt {1+{\bigg (}{\frac {dy}{dx}}{\bigg )}^{2}}}~dx.$
2. The surface area  $S$  of a function  $y=f(x)$  rotated about the  $y$-axis is given by
$S=\int 2\pi x\,ds,$  where $ds={\sqrt {1+{\bigg (}{\frac {dy}{dx}}{\bigg )}^{2}}}.$

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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