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son4ous [18]
3 years ago
10

Find the exact length of the curve. x = 9 + 9t2, y = 6 + 6t3, 0 ≤ t ≤ 4

Mathematics
1 answer:
andrew11 [14]3 years ago
3 0

ANSWER

=102 \sqrt{ 17}

EXPLANATION

The equation of the curve is:

x = 9 + 9 {t}^{2}

y = 6 + 6 {t}^{3}

We differentiate to obtain:

\frac{dx}{dt}  = 18t

\frac{dy}{dt}  = 18t

The length of the arc

l =  \int _0 ^{4}  \sqrt{  {( \frac{dx}{dt} )}^{2} + {( \frac{dy}{dt} )}^{2} } dt

This implies that:

=  \int _0 ^{4}  \sqrt{  {( 18t )}^{2} + {(  {18 {t}^{2} } )}^{2} } dt

= \int _0 ^{4} 18t \sqrt{  1+ { {18 {t}^{2} } }} dt

=102 \sqrt{ 17}

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