This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Volume of Solid of Revolution”.
1. The volume of solid of revolution when rotated along x-axis is given as _____________
a) \(\int_a^b πy^2 dx \)
b) \(\int_a^b πy^2 dy \)
c) \(\int_a^b πx^2 dx \)
d) \(\int_a^b πx^2 dy \)
View Answer
Explanation: Volume is generated when a 2d surface is revolved along its axis. When revolved along x-axis, the volume is given as \(\int_a^b πy^2 dx \).
2. The volume of solid of revolution when rotated along y-axis is given as ________
a) \(\int_a^b πy^2 dx \)
b) \(\int_a^b πy^2 dy \)
c) \(\int_a^b πx^2 dx \)
d) \(\int_a^b πx^2 dy \)
View Answer
Explanation: Volume is generated when a 2d surface is revolved along its axis. When revolved along y-axis, the volume is given as \(\int_a^b πx^2 dy \).
3. What is the volume generated when the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is revolved about its minor axis?
a) 4 ab cubic units
b) \(\frac{4}{3} a^2 b \) cubic units
c) \(\frac{4}{3} ab \) cubic units
d) 4 cubic units
View Answer
Explanation: y- axis is the minor axis. \(x^2 = \frac{a^2}{b^2} (b^2 – y^2)\)
\(V = \int_a^b πx^2 dy\)
\(= \int_{-b}^b π \frac{a^2}{b^2} (b^2 – y^2) \,dy \)
\(= 2π \frac{a^2}{b^2} \Big(b^2 y- \frac{y^3}{3}\Big)_0^b \)
\(= 2π \frac{a^2}{b^2} (b^3- \frac{b^3}{3}) \)
\(= \frac{4}{3} a^2 b \) cubic units.
4. What is the volume generated when the region surrounded by y = \(\sqrt{x}\), y = 2 and y = 0 is revolved about y – axis?
a) 32π cubic units
b) \(\frac{32}{5} \) cubic units
c) \(\frac{32π}{5}\) cubic units
d) \(\frac{5π}{32} \) cubic units
View Answer
Explanation: Limits for y -> 0,2 x = y2
\(Volume = \int_a^b πx^2 dy\)
\( = \int_0^2 πy^4 dy\)
\( = \Big[\frac{πy^5}{5}\Big]_0^2\)
\( = \frac{32π}{5}\) cubic units.
5. What is the volume of the sphere of radius ‘a’?
a) \(\frac{4}{3} πa \)
b) 4πa
c) \(\frac{4}{3} πa^2 \)
d) \(\frac{4}{3} πa^3 \)
View Answer
Explanation: The equation of a circle is x2 + y2 = a2
When it is revolved about x-axis, the volume is given as
\(V = 2 \int_a^b πy^2 dy\)
\(= 2 \int_0^a π(a^2-x^2) dx\)
\(= 2π \Big(a^2 x – \frac{x^3}{3}\Big)_0^a\)
\(= \frac{4}{3} πa^3.\)
6. Gabriel’s horn is formed when the curve ____________ is revolved around x-axis for x≥1.
a) y = x
b) y = 1
c) y = 0
d) y = 1/x
View Answer
Explanation: Gabriel’s horn or Torricelli’s Trumpet is a famous paradox. It has a finite volume but infinite surface area.
Sanfoundry Global Education & Learning Series – Differential and Integral Calculus.
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