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.

Sanfoundry Global Education & Learning Series – Differential and Integral Calculus.

To practice all areas of Differential and Integral Calculus, here is complete set of 1000+ Multiple Choice Questions and Answers.

If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]

Subscribe to our Newsletters (Subject-wise). Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

Youtube | Telegram | LinkedIn | Instagram | Facebook | Twitter | Pinterest
Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

Subscribe to his free Masterclasses at Youtube & discussions at Telegram SanfoundryClasses.