Differential and Integral Calculus Questions and Answers – Volume of Solid of Revolution

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) $$\int_a^b πx^2 dy$$

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) $$\int_a^b πx^2 dy$$

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

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

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) $$\frac{4}{3} πa^3$$

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

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.