Volumes and Surfaces of Revolution-Previous year questions

Previous Year Questions on Volumes and Surfaces of Revolution

Volumes and Surfaces of Revolution is a crucial topic in Mathematics, forming the foundation of 3D geometry and applications of Integral Calculus. Historically, the study of revolved shapes has enabled advancements in engineering, architecture, and physics. Frequently featured in Mathematics questions, this topic tests understanding of mathematical concepts and their practical use.

Vidyasagar University

2023-24 (NEP)

Find the volume of the solid generated by the revolution of the area enclosed by the astroid \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} \) about the x-axis. [2]

2023-24 (CBCS)

The volume of the solid generated by \(y=\frac{1}{x} \), bounded by \(y=0, x=2,x=b~(0\lt b \lt 2) \) about x-axis is \(3\). Find the value of \(b\). [2]

2022-23 (CBCS)

No Questions

2021-22 (CBCS)

No Questions

2020-21 (CBCS)

No Questions

2019-20 (CBCS)

No Questions

2018-19 (CBCS)

Prove that the surface of the solid obtained by revolving the ellipse \begin{align*} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \end{align*} round its minor axis is \begin{align*} 2\pi a^{2}\big[1+\frac{1-e^{2}}{2e}\log \big(\frac{1+e}{1-e} \big) \big] \end{align*} where \(b^{2}=a^{2}(1-e^{2})\). [5]

2017-18 (CBCS)

The voloume of the solid generated by the revolution of the curve \(y=\frac{1}{x}\), bounded by \(y=0,~x=2,~x=b~(0\lt b \lt 2)\) about x-axis is \(3\). Find the value of \( b\).
Find the area of the surface generated by revolving the curves \(x=\cos {t},~y=2+\sin {t}\), \(0\leq t\leq 2\pi\) about x-axis. [5]

FAQs

What are Volumes and Surfaces of Revolution?
These are volumes and surface areas obtained when a curve is revolved around an axis using methods from Integral Calculus.
Where are Volumes and Surfaces of Revolution applied?
They are used in engineering, architecture, and 3D modeling in physics.
What is the significance of this topic in universities?
It is a common subject in Mathematics questions, testing advanced problem-solving skills.
What methods are used to calculate these volumes?
Definite integration and formulae based on axis of revolution are used.
What types of curves are studied in this topic?
Polynomial, trigonometric, and exponential curves are commonly analyzed.
How does this topic relate to Integral Calculus?
It is an application of Integral Calculus in 3D geometry.
What is the shell method in this context?
It is a technique for finding volumes by integrating cylindrical shells.
How can students practice this topic effectively?
By solving Mathematics questions and exploring practical problems.
Are these techniques included in competitive exams?
Yes, they are integral to many advanced-level Mathematics and engineering tests.
What is the role of symmetry in solving these problems?
Symmetry simplifies calculations by reducing the complexity of definite integration.