Integral Calculus and Differential Equations Test

Result

Integral Calculus and Differential Equations Test - 7

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Weekly Quiz Competition

Question 1
2 / -0.83

If A is the region bounded by the parabolas y² = 4x and x² = 4y then is equal to

A
48/5

B
36/5

C
32/5

D
None of these

Solution
Question 2
2 / -0.83

The area of the region bounded by the curves x² + y² = a² and x + y = a in the first quadrant is given by

A

B

C

D
None of these

Solution

The curves are

x² + y² = a² ...(i)

and x + y = a ...(ii)

The curves (i) and (ii) intersect at A (a, 0) and B (0,a)
Question 3
2 / -0.83

The area bounded by the curves y = 2 √x , y = -x , x = 1 and x = 4 is given by

A
25

B
33/2

C
47/4

D
101/6

Solution

The given equations of the curves are
Question 4
2 / -0.83

The area of the cardioid r = a (1 + cos θ) is given by

A

B

C

D

Solution

The equation of the cardioid is

r = a (1 + cos θ) .... (i)

If a figure is drawn then from fig. the required area is
Question 5
2 / -0.83

The area bounded by the curve r = θcos θand the lines θ= 0 and θ= π/2 is given by

A

B

C

D
none of these

Solution

The equation of the given curve is

r = θcos θ...(i)

The required area
Question 6
2 / -0.83

The area of the lemniscate r² = a² cos2 θis given by

A

B

C

D

Solution

If a figure is drawn then from fig. the required area is
Question 7
2 / -0.83

The area of the region bounded by the curve y(x² + 2) = 3x and 4y = x² is given by

A

B

C

D

Solution

The equations of given curves are

y(x² + 2) = 3x ....(i) and 4y = x² ....(ii)

The curve (i) and (ii) intersect at A (2, 1).
If a figure is drawn then from fig. the required area is
Question 8
2 / -0.83

The volume of the cylinder x² + y² = a² bounded below by z = 0 and bounded above by z = h is given by

A
πah

B
πa² h

C
1/3 πa ³ h

D
None of these

Solution

The equation of the cylinder is x² + y² = a²

The equation of surface CDE is z = h

If a figure is drawn then from fig. the required area is
Question 9
2 / -0.83

A
(e - 1)³

B
3/2(e - 1)

C
(e - 1)²

D
none of these

Solution
Question 10
2 / -0.83

The area of the region, enclosed by the circle x² + y² = 2 which is not common to the region bounded by the parabola y² = x and the straight line y = x is

A
1/3 (12 π- 1)

B
1/6 (12 π- 1)

C
1/3 (6 π- 1)

D
1/6 (24 π- 1)

Solution

Required area = area of the circle –area bounded by given line and parabola

Required area = πr² –

= 2 π –(1/6)
= (1/6)(12 π –1)