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Let $\displaystyle {C}$ be the curve $\displaystyle {y}={4}\sqrt{{{x}}}$ for $\displaystyle {1.8}\le{x}\le{4.2}$ .

Find the surface area of revolution about the $\displaystyle {x}$ -axis of $\displaystyle {C}$ .

3dsurface

Surface area $\displaystyle ={\int_{{1.8}}^{{4.2}}}{f{{\left({x}\right)}}}\ {\left.{d}{x}\right.}$ where

$\displaystyle {f{{\left({x}\right)}}}=$

Write your answer in this format: $\displaystyle {2}\pi\cdot{4}\sqrt{{{x}+\frac{{4}^{{2}}}{{4}}}}$

Now integrate to find surface area

Surface area $\displaystyle ={\int_{{1.8}}^{{4.2}}}{f{{\left({x}\right)}}}\ {\left.{d}{x}\right.}=$

To find the surface area of revolution, recall the formula for surface area when rotating a curve around the

\displaystyle {x}

-axis. Consider how to express the derivative

\displaystyle \frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}

for the given function, and how it fits into the surface area formula. Remember to set up your integral carefully, paying attention to the given bounds.

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