W04 Regular

Due date: Sunday 2/8, 11:59pm

Arc length

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Arc length - tricky integration

Find the arc length of the curve $y=e^x$ for $x\in [\mathrm{0,1}/2]$.

(Hint: the integral can be done using either: (i) $u$-sub then trig sub, or (ii) ‘rationalization’ then partial fractions.)

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Solution

Solutions---0080-03

Surface areas of revolutions - thin bands

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Surface area: cone

A cone may be described as the surface of revolution of a ray emanating from the origin, revolved around the $x$-axis.

Let $f(x)=mx$ for some $m>0$. Find the surface area of the cone given by revolving the graph of $f$ around the $x$-axis over $x\in [0,h]$.

Now calculate this area using geometry, and verify that the two methods give the same formula. (Hint: ‘unroll’ the cone into a sector.)

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Solution

Solutions---0090-02

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Surface area: parabolic reflector

A parabolic reflector is given by rotating the curve $y=x^2$ around the $y$-axis for $x\in [\mathrm{0,2}]$.

What is the surface area of this reflector?

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Solution

Solutions---0090-03