Double Integral MCQ Quiz - Objective Question with Answer for Double Integral - Download Free PDF

Explanation:

= 

= 

= 

= 

Thus, 

∴ a = 25

Explanation:

Given curve,

At x = 0 ⇒ y = 0

So the diagram for the given curve is

Required area

Put

 = 16 

Hence Option(2) is the correct answer.

Concept:

Surface area of portion of plane in cylinder is:

S= 

Explanation: 

Given y + 2z = 2 and

Therefore Surface area of portion of plane in cylinder is:

S= 

Hence Option(2) is the correct answer.

Explanation:

= 

=  (as )

= 0 (as  if f(x) is odd function)

Option (3) is true.

Calculation:

Given, f(x, y) =   du dv.

Let u - x = r cosθ and v - y = r sinθ 

⇒ du dv = r dr dθ

and, (u - x)² + (v - y)² ≤ 1 represents a circle of radius, r ≤ 1.

∴    du dv

= 

= 

=2π

= 2π

= 2π

= 2π[(- e^-1 - e^-1) - (0 - 1)]

= 2π(1 − 2e−1) 

∴   f(n, n2) = 2π(1 − 2e−1) 

(4) is correct

Given:

0 ≤ y ≤ x² (this is represented by vertical strip)

And x varies from 0 to 1.

Now if we change the order of integration, we have to draw a horizon strip.

After changing the order of Integration

And, 0 ≤ y ≤ 1

∴ 

The equation of line in intercept form is given by

⇒ 

⇒ 

Given,

and x² + y² ≤ 4

Putting x = r.cosθ, y = r.sinθ and dx.dy = r.dr.dθ

The given curves are y = x and y = x²

Solving the equations, we get

x  = 0, x = 1

Alternate Solution:

Concept:

Area of a region can be calculated by:

Calculation:

Solving equation y = x² and we get y = x

We get intersection points i.e. (0,0) and (1,1)

Area of the region:

Changing the order and take vertical strip, we get:

I = (1 – (-1))

I = 2

Explanation:

We have the integration given as,

Placing the limits we get,

Hence the required value of integration will be .

Concept:

Ellipse

Length major axis of ellipse = 2a

Length of minor axis of ellipse = 2b

Area

Calculation:

.

For the first quadrant, take a vertical strip as shown. Here, y coordinate varies from 0 to .

Also, the x-coordinate varies from 0 to a

∴ Area

∴ The total area of ellipse  = πab units

Important Points

1) x² + y² = a² ; Represents a circle centred at (0, 0) and radius ‘a’ units.

2) ; Represents Ellipse with major axis length 2a and minor axis length 2b and vertex at (0,0)

3) ; Equation of Hyperbola .

4) ; Equation of Rectangular Hyperbola.

Calculation

Given equations are: y² = 9x, x – y + 2 = 0

By solving the above two equations,

The point of intersection of the two curves are: (1, 3) and (4, 6)

Now, the graph is shown below.

By considering the horizontal strip,

The limits of y are:  3 to 6

The limits of x are: (y – 2) to y²/9

Now, the required area is 

Concept:

Gauss divergence theorem

Calculation:

2 × 36 π = 72 π = 226.19