Integral Calculus Questions and Answers – Rectification in Polar and Parametric Forms


This set of Differential and Integral Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Rectification in Polar and Parametric Forms”.

1. Find the length of the curve given by the equation.
a) \(\frac{3a}{2}\)
b) \(\frac{-7a}{2}\)
c) \(\frac{-3a}{4}\)
d) \(\frac{-3a}{2}\)
View Answer

Answer: d
Explanation: We know that,
Differentiating on both sides
\(\frac{2}{3} y^{\frac{2}{3}-1}= \frac{-2}{3} x^{\frac{2}{3}-1}\)
\(\frac{dy}{dx} = -\frac{y}{x}^{\frac{1}{3}}\)
\((\frac{dy}{dx})^2 = (\frac{y}{x})^{\frac{1}{3}}\)
Substituting from the original equation-
\(S=\int_{a}^{0}(\frac{a}{x})^{\frac{1}{3}} dx \)
Thus, length of the given curve is \(\frac{-3a}{2}\).

2. Find the length of one arc of the given cycloid.


a) a
b) 4a
c) 8a
d) 2a
View Answer

Answer: c
Explanation: We know that
\((\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2=a^2(1-cos\theta)^2+a^2 sin^2\theta\)
\((\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2=4a^2 sin^2\frac{\theta}{2}\)
\(s=\int_{0}^{2}\pi\sqrt{4a^2 sin^2\frac{\theta}{2}} d\theta\)
On solving the given integral, we get
Thus length of one arc of the given cycloid is 8a.

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