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

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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.