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All questions of Integral Calculus for Mathematics Exam
The area bounded by the curves y2 = 9x, x - y + 2 = 0 is given by:- a)1
- b)1/2
- c)3/2
- d)5/4
Correct answer is option 'B'. Can you explain this answer?
The area bounded by the curves y2 = 9x, x - y + 2 = 0 is given by:
a)
1
b)
1/2
c)
3/2
d)
5/4
|
Veda Institute answered |
The equations of the given curves are
y2 = 9x ...(I)
x - y + 2 = 0 ...(II)
The curves (i) and (ii) intersect at A(1, 3) and B(4, 6) Hence The required area
y2 = 9x ...(I)
x - y + 2 = 0 ...(II)
The curves (i) and (ii) intersect at A(1, 3) and B(4, 6) Hence The required area
The volume of the solid generated by the revolution of the cardioid r = a(1 + cos θ) about the initial line is given by- a)2πa3
- b)2/3πa3
- c)4/3πa3
- d)8/3πa3
Correct answer is option 'D'. Can you explain this answer?
The volume of the solid generated by the revolution of the cardioid r = a(1 + cos θ) about the initial line is given by
a)
2πa3
b)
2/3πa3
c)
4/3πa3
d)
8/3πa3
|
Chirag Verma answered |
The curve is r = a (1 + cos θ)
Required volume
Required volume
then f(3) + f(1) equals:
The length of the arc of the curve 6xy = x4 + 3 from x = 1 to x = 2 is:- a)13/12 units
- b)17/12 units
- c)19/12 units
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
The length of the arc of the curve 6xy = x4 + 3 from x = 1 to x = 2 is:
a)
13/12 units
b)
17/12 units
c)
19/12 units
d)
None of these
|
|
Chirag Verma answered |
The given curve is
6xy = x4 + 3
Thus Length of the arc between x
= 1 and x = 2 is:
6xy = x4 + 3
Thus Length of the arc between x
= 1 and x = 2 is:
x dx is equal to- a)8/15
- b)4/15
- c)1/5
- d)0
Correct answer is option 'D'. Can you explain this answer?
x dx is equal toa)
8/15
b)
4/15
c)
1/5
d)
0
|
Sakshi Jain answered |
Sinx is an odd function so from property of integration( i.e when limit is from -a to a) so its value is zero
The surface area of the segment of a sphere of radius a and height h is given by:- a)2πh
- b)2πah
- c)2πa2h
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
The surface area of the segment of a sphere of radius a and height h is given by:
a)
2πh
b)
2πah
c)
2πa2h
d)
None of these
|
|
Chirag Verma answered |
Let the sphere be generated by the revolution about the x - axis of the circle
x2 + y2=a2 ...(i)
Let OA =a, OC = b and OB = b + h
Hence The required surface
x2 + y2=a2 ...(i)
Let OA =a, OC = b and OB = b + h
Hence The required surface
The intrinsic equations of the cardioids r = a (1 - cos θ) and r = a(1 + cos θ) measured from the pole are:- a)
- b)
- c)
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
The intrinsic equations of the cardioids r = a (1 - cos θ) and r = a(1 + cos θ) measured from the pole are:
a)
b)
c)
d)
None of these
|
|
Veda Institute answered |
The equations of given cardioids are
r= a (l -cos θ) ...(i)
r= a (1 + cos θ) ...(ii)
Intrinsic equation for cardioid (i):
...(iii)
r= a (l -cos θ) ...(i)
r= a (1 + cos θ) ...(ii)
Intrinsic equation for cardioid (i):
...(iii)
is given by:
dt, then the derivative of f(x) with respect to x is:- a)
- b)
- c)
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
dt, then the derivative of f(x) with respect to x is:a)
b)
c)
d)
None of these
|
Vaani Jain answered |
Apply Leibniz's rule for differentiation under the integral sign,
- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
a)
b)
c)
d)
|
|
Veda Institute answered |
Given that
Hence
Hence
equals:
The line y = x + 1 is revolved about x-axis. The volume of solid of revolution formed by revolving the area covered by the given curve, x-axis and lines x = 0, x = 2 is:- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
The line y = x + 1 is revolved about x-axis. The volume of solid of revolution formed by revolving the area covered by the given curve, x-axis and lines x = 0, x = 2 is:
a)
b)
c)
d)
|
|
Chirag Verma answered |
The given curve is y = x + 1. This represents a straight line
Hence The required volume
Hence The required volume
is equal to
The moment of inertia of a hollow sphere about a diameter is:
- a)Ma2
- b)
- c)
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
The moment of inertia of a hollow sphere about a diameter is:
a)
Ma2
b)
c)
d)
None of these
|
|
Veda Institute answered |
Moment of Inertia of a Hollow Sphere about the Diameter
Suppose the mass of a hollow sphere is M, ρ is the density, inner radius R2 and outer radius R1,
∴ M = 4/3π(R13 − R23)ρ
Moment of inertia of a hollow sphere (I) = Moment of inertia of a solid sphere of radius R1 - Moment of inertia of a solid sphere of radius R2
is equal to:The value of 
is equal to: - a)0
- b)π
- c)∞
- d)8π
Correct answer is option 'A'. Can you explain this answer?
The value of is equal to:
is equal to: a)
0
b)
π
c)
∞
d)
8π
|
|
Chirag Verma answered |
sin(4x)
=> sin(2x+2x)
=> sin(2x).cos(2x)+cos(2x).sin(2x)
=> 2[sin(x).cos(x)].cos(2x)+2[sin(x).cos(x)].cos(2x)
=> sin(4x)sin(x)=>2cos(x).cos(2x)+2cos(x).cos(2x)
Our integral is now reduced to 4cos(x).cos(2x)
=> sin(2x+2x)
=> sin(2x).cos(2x)+cos(2x).sin(2x)
=> 2[sin(x).cos(x)].cos(2x)+2[sin(x).cos(x)].cos(2x)
=> sin(4x)sin(x)=>2cos(x).cos(2x)+2cos(x).cos(2x)
Our integral is now reduced to 4cos(x).cos(2x)
I = ∫(0 to π) 4cos(x).cos(2x)
I = ∫(0 to π) 4cos(x).(1−2sin2(x))
I = ∫(0 to π) 4cos(x)−8sin2(x).cos(x))
I = ∫(0 to π)4cos(x) - ∫8sin2(x).cos(x))
∫(0 to π) cos(x)=sin(x)
∫(0 to π) sin2(x).cos(x))=sin3(x)3
I = [sin(x)+8sin3(x)3+c ](0 to π)
I = ∫(0 to π) 4cos(x).(1−2sin2(x))
I = ∫(0 to π) 4cos(x)−8sin2(x).cos(x))
I = ∫(0 to π)4cos(x) - ∫8sin2(x).cos(x))
∫(0 to π) cos(x)=sin(x)
∫(0 to π) sin2(x).cos(x))=sin3(x)3
I = [sin(x)+8sin3(x)3+c ](0 to π)
Putting limits, hence we get answer 0.
The area bounded by the parabola y2 = 4ax and its latus rectum is given by- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
The area bounded by the parabola y2 = 4ax and its latus rectum is given by
a)
b)
c)
d)
|
|
Chirag Verma answered |
The equation of the parabola is
y2 = 4 ax
We have to find the area OL'MLO
Hence Area OL'MLO = 2 • Area OMLO
y2 = 4 ax
We have to find the area OL'MLO
Hence Area OL'MLO = 2 • Area OMLO
The value of 
- a)25/33
- b)32/231
- c)64/231
- d)28/33
Correct answer is option 'C'. Can you explain this answer?
The value of
a)
25/33
b)
32/231
c)
64/231
d)
28/33
|
Ibrahim Parekh answered |
Take sin (x)=t
cos (x)dx=dt
cos (x)dx=dt
The segment o f the circle x2 + y2 = a2 cut off by the chord x = b (0 < b < a) revolves about the x-axis and generates the solid known as a segment of a sphere. The volume of this solid is:- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
The segment o f the circle x2 + y2 = a2 cut off by the chord x = b (0 < b < a) revolves about the x-axis and generates the solid known as a segment of a sphere. The volume of this solid is:
a)
b)
c)
d)
|
|
Chirag Verma answered |
The curve bounded by x2 + y2 = a2 and chord x = b revolved about x-axis, then
The required volume is given by
The required volume is given by
is equal to 
- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
a)
b)
c)
d)
|
|
Veda Institute answered |
Let us assume that 
Therefor
Therefor
The length of the complete cycloid x = a (θ + sin θ), y = a(1 - cos θ) is given by:- a)a
- b)4a
- c)8a
- d)32 a
Correct answer is option 'C'. Can you explain this answer?
The length of the complete cycloid x = a (θ + sin θ), y = a(1 - cos θ) is given by:
a)
a
b)
4a
c)
8a
d)
32 a
|
|
Chirag Verma answered |
The equations of the curve are
x = a(θ + sin θ), y = a(1 - cos θ)
Hence The entire length of the curve
x = a(θ + sin θ), y = a(1 - cos θ)
Hence The entire length of the curve
The volume of the solid generated by the revolution of r = 2a cos θ about the initial line is given by:- a)2/3πa3
- b)4/3πa3
- c)8/3πa3
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
The volume of the solid generated by the revolution of r = 2a cos θ about the initial line is given by:
a)
2/3πa3
b)
4/3πa3
c)
8/3πa3
d)
None of these
|
|
Chirag Verma answered |
The equation of the given curve is
r= 2a cos θ,
which is a circle with centre (a, 0) and radius a.
Hence The required volume
r= 2a cos θ,
which is a circle with centre (a, 0) and radius a.
Hence The required volume
xF(sin X) dx is equal to- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
xF(sin X) dx is equal toa)
b)
c)
d)
|
|
Chirag Verma answered |
...(i)
...(ii)Adding (i) and (ii), we get
Hence
The area bounded by the curve x2 = y and y2 = x is given by:- a)1
- b)1/3
- c)2/3
- d)3/2
Correct answer is option 'B'. Can you explain this answer?
The area bounded by the curve x2 = y and y2 = x is given by:
a)
1
b)
1/3
c)
2/3
d)
3/2
|
|
Chirag Verma answered |
Given equations of the curves are
y2 = x ...(i)
x2 = y ...(ii)
Solving these equations, we get
x = 0, x = 1
Hence required area OPAQO
y2 = x ...(i)
x2 = y ...(ii)
Solving these equations, we get
x = 0, x = 1
Hence required area OPAQO
The area enclosed by the curve r2 = a2 sinθ is given by:- a)a2
- b)2a2
- c)2a
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
The area enclosed by the curve r2 = a2 sinθ is given by:
a)
a2
b)
2a2
c)
2a
d)
None of these
|
|
Veda Institute answered |
The equation of the given curve is
r2 = a2sinθ
Required area
= Area of OBACO
= 2 x Area OBAO
r2 = a2sinθ
Required area
= Area of OBACO
= 2 x Area OBAO
The intrinsic equation of the cycloid x = a (θ + sin θ) , and y = a (1 - cos θ) is given by:- a)
- b)
- c)
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
The intrinsic equation of the cycloid x = a (θ + sin θ) , and y = a (1 - cos θ) is given by:
a)
b)
c)
d)
None of these
|
|
Veda Institute answered |
Given equations of cycloid are
Then
and
...(ii)
Hence
So,
Implies
Hence From (iii), we ge
Then
and
...(ii)Hence
So,
Implies
Hence From (iii), we ge
The intrinsic equation of the curve p = r sin α is given by: - a)
- b)
- c)
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
The intrinsic equation of the curve p = r sin α is given by:
a)
b)
c)
d)
None of these
|
|
Veda Institute answered |
Given curve is p = r sin α ....(i)
Hence
or,
Hence
or,
The length of the arc of the parabola x2 = 4ay from the vertex to the extremity of the latus rectum is given by:- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
The length of the arc of the parabola x2 = 4ay from the vertex to the extremity of the latus rectum is given by:
a)
b)
c)
d)
|
|
Veda Institute answered |
Given curve is x2 = 4ay
Differentiating, w.r. to y, we get
implies
implies
Hence Length of the arc between vertex and extremity of latus rectum
Differentiating, w.r. to y, we get
implies
implies
Hence Length of the arc between vertex and extremity of latus rectum
The length of the are of the curve
x sin θ + y cos θ = f' (θ),
x cos θ - y cos θ = f " (θ)
is given by:- a)f (θ) + f' (θ) + constant
- b)f (θ) + f '' (θ) + constant
- c)f ' (θ) + f ''(θ) + constant
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
The length of the are of the curve
x sin θ + y cos θ = f' (θ),
x cos θ - y cos θ = f " (θ)
is given by:
x sin θ + y cos θ = f' (θ),
x cos θ - y cos θ = f " (θ)
is given by:
a)
f (θ) + f' (θ) + constant
b)
f (θ) + f '' (θ) + constant
c)
f ' (θ) + f ''(θ) + constant
d)
None of these
|
|
Chirag Verma answered |
The equations of the given curve are
x sin θ + y cos θ = f' (θ),
x cos θ - y cos θ = f " (θ)
Solving these equations, we get
Differentiating these w.r. to θ, we get
Hence the required length is given by
x sin θ + y cos θ = f' (θ),
x cos θ - y cos θ = f " (θ)
Solving these equations, we get
Differentiating these w.r. to θ, we get
Hence the required length is given by
sec x(1 + tan x) dx is equal to
x sec2 (x2 +1) tan (x2 +1) dx is equal to
is equal to
The length of the cycloid with parametric equation x(t) = (t + sin t), y(t) = (1 - cos t) between (0, 0) and (π, 2) is:- a)4 unit
- b)4π unit
- c)2 unit
- d)π/2 unit
Correct answer is option 'A'. Can you explain this answer?
The length of the cycloid with parametric equation x(t) = (t + sin t), y(t) = (1 - cos t) between (0, 0) and (π, 2) is:
a)
4 unit
b)
4π unit
c)
2 unit
d)
π/2 unit
|
|
Chirag Verma answered |
The given curve is
x(t) = (t + sin t), y(t) = (1 - cos t)
x(t) = (t + sin t), y(t) = (1 - cos t)
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