The total length of the curve r-asin3 -3 is ab- The value of b is Curve is defined for 0- 3-

F(θ )=asin^3 ( θ3 ) f'(θ )=asin^2 ( θ3 )cos ( θ3 ) S=∫_0^3π√( [ asin^3 ( θ3 ) ]^2+ [ asin^2 ( θ3 ) cos ( θ3 ) ]^2dθ) =∫_0^3πasin^2 ( θ3 )dθ=a2∫_0^3π1 cos ( 2 ...

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Standard X

Mathematics

Trigonometric Identities

The total len...

Question

$The total length of the curve r = a {sin}^{3} (\frac{θ}{3}) is a b π . The value of b is _____ (Curve is defined for 0 \leq θ \leq 3 π)$
1.5

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Solution

The correct option is A 1.5
$f (θ) = a s i n^{3} (\frac{θ}{3})$
$f^{'} (θ) = a s i n^{2} (\frac{θ}{3}) c o s (\frac{θ}{3})$

$S = \int_{0}^{3 π} \sqrt{{[a s i n^{3} (\frac{θ}{3})]}^{2} + {[a s i n^{2} (\frac{θ}{3}) c o s (\frac{θ}{3})]}^{2} d θ}$

$= \int_{0}^{3 π} a s i n^{2} (\frac{θ}{3}) d θ = \frac{a}{2} \int_{0}^{3 π} 1 - c o s (\frac{2 θ}{3}) d θ$

$= \frac{a}{2} [θ {∣ ∣ ∣_{0}^{3 π} - \frac{3}{2} s i n (\frac{2 θ}{3}) ∣ ∣ ∣}_{0}^{3 π}] = \frac{a}{2} (3 π)$

$= a b π$

$\Rightarrow b = 1.5$

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The total length of the curve r=asin3(θ3) is abπ. The value of b is _____ (Curve is defined for 0≤θ≤3π)1.5

The correct option is A 1.5f(θ)=asin3(θ3) f′(θ)=asin2(θ3)cos(θ3) S=∫3π0√[asin3(θ3)]2+[asin2(θ3)cos(θ3)]2dθ =∫3π0asin2(θ3)dθ=a2∫3π01−cos(2θ3)dθ =a2[θ∣∣∣3π0−32sin(2θ3)∣∣∣3π0]=a2(3π) =abπ ⇒ b=1.5

$The total length of the curve r = a {sin}^{3} (\frac{θ}{3}) is a b π . The value of b is _____ (Curve is defined for 0 \leq θ \leq 3 π)$
1.5