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

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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.
\(x^{\frac{2}{3}}+y^\frac{2}{3}=a^\frac{2}{3}\)
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 \(\frac{-3a}{2}\).
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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
\(s=\int_{\theta1}^{\theta2}\sqrt{(\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2}\)
\(\frac{dx}{d\theta}=a(1-cos\theta)\)
\(\frac{dy}{d\theta}=a(-sin\theta)\)
\((\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2=a^2(1-cos\theta)^2+a^2 sin^2\theta\)
\((\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2=4a^2 sin^2\frac{\theta}{2}\)
\(s=\int_{0}^{2}\pi\sqrt{4a^2 sin^2\frac{\theta}{2}} d\theta\)
On solving the given integral, we get
s=8a
Thus length of one arc of the given cycloid is 8a.

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

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

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn