This set of Differential and Integral Calculus Interview Questions and Answers for freshers 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 = y

^{2}

\(Volume = \int_a^b πx^2 dy\)

\( = \int_0^2 πy^4 dy\)

\( = \Big[\frac{πy^5}{5}\Big]_0^2\)

\( = \frac{32π}{5}\) 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 x

^{2}+ y

^{2}= a

^{2}

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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