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QUESTION: 1

The value of dydx by changing the order of integration is

Solution:

QUESTION: 2

The volume of ellipsoide is

Solution:

QUESTION: 3

The area bounded by the curve y = ψ(x), x-axis and the lines x = l , x = m(l <m ) is given by

Solution:

QUESTION: 4

The volume of an object expressed in spherical coordinates is given by sin φ dr dφ dθ. The value of the integral is

Solution:

QUESTION: 5

dx dy is equal to

Solution:

QUESTION: 6

Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integral is

Solution:

QUESTION: 7

The area bounded by the parabola y^{2} = 4ax and straight line x + y = 3a is

Solution:

QUESTION: 8

Consider the shaded triangular region P shown in the figure, what is the value of ?

Solution:

QUESTION: 9

Changing the order of the integration in the double leads to

. What is q?

Solution:

QUESTION: 10

To evaluate over the region A bounded by the curve r = r_{1}, r = r_{2} and the straight lines θ = θ_{1}, θ = θ_{2}, we first integrate

Solution:

QUESTION: 11

To change Cartesian plane (x, y, z) to spherical polar coordinates (r, θ, φ)

Solution:

To convert a point from Cartesian coordinates to spherical coordinates, use equations

r^{2} = x^{2} + y^{2} + z^{2} , tanθ = y/x, φ = arccos(z/√(x^{2}+y^{2}+z^{2}) )

QUESTION: 12

The double integral ,where D is bounded by y = x and y^{2} = 4x is equal to

Solution:

QUESTION: 13

What is the total mass of cube between the limits 0≤ x≤ 1, 0≤ y≤ 1, 0 ≤z ≤1 at any point given by xyz?

Solution:

QUESTION: 14

Area bounded by the curves y^{2} = x^{3} and x^{2} = y^{3} is

Solution:

QUESTION: 15

The value of (x+y+z) dzdydx is

Solution:

QUESTION: 16

By changing the order of integration in the value is

Solution:

QUESTION: 17

By the change of variable x(u, v) = uv, y(u, v) = u/v is double integral, the integrand f(x, y) change to . Then, φ(u,v) is

Solution:

QUESTION: 18

The volume of the tetrahedron bounded by the plane and the co-ordinate planes is equal to

Solution:

Here

Let u = x/a, v = y/b , w = z/c

Then dx = a du, dy = b dv, dz = c dw

So, Required volume

V = abc du dv dw

where u + v + w ≤ 1, u ,v ,w ≥ 0

Thus

Hence, the correct answer is (c)

QUESTION: 19

The value of integral dxdy is

Solution:

I = dxdy

Putting x^{2 }+ y^{2} = r^{2}, we get

Limits of r = 0 to ∞

and 0 = θ to π/2

Now, Putting r^{2} = t, we get

QUESTION: 20

dxdy is

Solution:

sin (x+y) dxdy

Firstly integrating w.r.t. x, we get

Now , integrating w.r.t. y, we get,

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