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 )
Answer: a
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 )
Answer: d
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
Answer: b
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
Answer: c
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 )
Answer: d
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
Answer: d
Explanation: Gabriel’s horn or Torricelli’s Trumpet is a famous paradox. It has a finite volume but infinite surface area.
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