Application Of Integrals

If and Vis the region bounded by the surface x=0,y=0,x=2,y=4,z=x²,z=2, then value of is

The volume generated by the revolution of the cardioid about its axis is:

If and C is the bounding curve of the hemisphere x²+y²+z²=9,z>0, oriented in the positive direction, then value of is

Given below are two statements
Statement I: Draw back in Lagrange's method of undetermined multipliers is that nature of stationary point cannot be determined
Statement II: convergent
In the light of the above statements, choose the correct answer from the options given below

If , where R is the region bounded by x = 0, x = 2, y = x, y = x+2, then is equal to:

The surface area of the cylinder inside the cylinder is:

Integrating factors of the equation y (2xy + e^x) dx - e^x dy = 0 is :

The orthogonal trajectory of the cardioid r = a(1+cos θ), a being the parameter is:

Given below are two statements:
Statement I : If x²y" - 2xy' - 4y = x⁴, then
Statement II: If (D²-8D+15) y = 0, then auxiliary equation has equal roots.
In the light of the above statements, choose the correct answer from the options given below

If , then the line integral from (0.0, 0) to (1, 1.1), along the curve C ;x=t, y=t². z=t³ is:

A.
B.
C. grad is a vector normal to the surface ∅ = C
D. ∇∅ is the maximum rate of change of ∅
Choose the correct answer from the options given below

Given below are two statements:
Statement I: If ø (x, y), Ψ(x, y), and are continuous functions over region R bounded by simple closed curve C in xy-plane, then
Statement II: be a vector point function and volume V enclosed by a closed surface, then volume integral
In the light of the above statements, choose the most appropriate answer from the options given below

Match List I with List II

List I		List II
A.	Φ=y2,∇Φ at (1, 1, 1)	I.
B.	Φ=x,∇Φ at (1, -1, 2)	II.
C.	Φ=2x3,∇Φ at (0, 1, 2)	III.
D.	Φ=3z2,∇Φ at (1, 2, -1)	IV.

Choose the correct answer from the options given below:

A. vector is said to be solenoidal if div
B. Vector is said to be irrotational if curl
C. If represents the force acting on a particle along are AB, then total work done
D. Volume integral
choose the correct answer from the options given below