01
Arc length - reversed
and roles Find the arc length of the curve that satisfies the equation
over .
Solution
01
(1) Integral formula for arclength:
(2) Work out integrand:
(3) Integrate:
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02
Arc length - tricky algebra
Find the arc length of the curve
for . (Hint: expand under the root, then simplify, then factor; now it’s a square and the root disappears.)
Solution
03
(1) Integral formula for arclength:
(2) Work out integrand:
Therefore, the integrand:
(3) Integrate:
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03
Arc length - tricky integration
Find the arc length of the curve
for . (Hint: the integral can be done using either: (i)
-sub then trig sub, or (ii) ‘rationalization’ then partial fractions.)
Solution
04
(1) Integral formula for arclength:
(2) Perform
-sub with and so and also: Now transform the integral to
:
(3) Integrate: partial fraction decomposition:
Number degree not lower → long division first:
Write general PFD formula:
Solve for
and . Cross multiply:
(4) Evaluate integral:
Note A: Instead of this
-sub and partial fractions, one can set and obtain . Then trig sub with leads to (eventually) the same final answer. Note B: This answer is sufficient. It is not necessary to simplify as in the last step.
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