Integral Calculus NAT Level - 1 - Free MCQ Test with solutions for Physics

The points of intersection are

Total Area = 2(Area of I + Area of II)

Area of Region I

Area of Region II


Area of Region 

The correct answer is: -0.75

As, it contains only even powers of x, hence, it is symmetrical about y-axis
Total Area = 2 × area OAB

The correct answer is: 3.142

The curve is
x²y² = a²(y² – x²)
(1) Symmetry about both the axes at even powers of x and y occur.
(2) Asymptotes are x = ±a

The correct answer is: 4

Hence, the Area bounded is given as

As the curve is symmetrical.
Total Area = 3 × Area of loop OA




 

= 12.571
The correct answer is: 12.571 

Given curves are

Solving the two equations,

Here Required area will be

For Region I

For Region II


 

Hence, λ = 2.142

The correct answer is: 2.142

This can be done by two ways:
Ist Way:


IInd Way :
Putting x = 0 and y = 0, we find that the given curve meets y and x-axes in (0, 1) and (1, 0) respectively.

The correct answer is: 0.167

Points of intersection

Notice that D is the region such that for each fixed angle θ between  varies from 1 + cos θ (cardioid) to 3 cos θ (circle).

The correct answer is: 8

Changing to polar coordinates, we get

Putting r²=t

2r dr = dt

The correct answer is: 1

The region of integration A can be covered by radial strips whose ends are at r = 0, r = a (1 + cos θ). The strips like between θ = 0 and θ = π.

Thus,

The correct answer is: 1.333

The region of integration is

Now, changing to polar coordinates, we get

The correct answer is: 0.0327