Differential and Integral Calculus Multiple Choice Questions on “Rectification in Polar and Parametric Forms”.
1. Find the length of the curve given by the equation.
(x^{frac{2}{3}}+y^frac{2}{3}=a^frac{2}{3})
a) (frac{3a}{2})
b) (frac{-7a}{2})
c) (frac{-3a}{4})
d) (frac{-3a}{2})
View Answer
Answer: d
Explanation: We know that,
S=(int_{x1}^{x2}sqrt{1+frac{dy}{dx}^2})
(y^frac{2}{3}=a^frac{2}{3}-x^frac{2}{3})
Differentiating on both sides
(frac{2}{3} y^{frac{2}{3}-1}= frac{-2}{3} x^{frac{2}{3}-1})
(frac{dy}{dx} = -frac{y}{x}^{frac{1}{3}})
((frac{dy}{dx})^2 = (frac{y}{x})^{frac{1}{3}})
(1+(frac{dy}{dx})^2=1+(frac{y}{x})^frac{2}{3})
Substituting from the original equation-
(1+(frac{dy}{dx})^2=(frac{a}{x})frac{2}{3})
(sqrt{1+frac{dy^2}{dx}}=(frac{a}{x})^{frac{1}{3}})
(S=int_{a}^{0}(frac{a}{x})^{frac{1}{3}} dx )
(s=frac{-3a}{2})
Thus, length of the given curve is (frac{-3a}{2}).
2. Find the length of one arc of the given cycloid.
x=a(θ-sinθ) y=a(1+cosθ)
a) a
b) 4a
c) 8a
d) 2a
View Answer
Answer: c
Explanation: We know that
(s=int_{theta1}^{theta2}sqrt{(frac{dx}{dtheta})^2+(frac{dy}{dtheta})^2})
(frac{dx}{dtheta}=a(1-costheta))
(frac{dy}{dtheta}=a(-sintheta))
((frac{dx}{dtheta})^2+(frac{dy}{dtheta})^2=a^2(1-costheta)^2+a^2 sin^2theta)
((frac{dx}{dtheta})^2+(frac{dy}{dtheta})^2=4a^2 sin^2frac{theta}{2})
(s=int_{0}^{2}pisqrt{4a^2 sin^2frac{theta}{2}} dtheta)
On solving the given integral, we get
s=8a
Thus length of one arc of the given cycloid is 8a.
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