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Integration and Its Applications Questions and Answers - Form 4 Calculus
Evaluate
∫
3
-
1
(
2
x
+
3
)
d
x
(2m 34s)
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1.
A particle moves along a straight line PQ. Its velocity v metres per second after t seconds is given by
v
=
t
2
-
3
t
+
5
. Its distance from p at the time t=1 is 6 metres . Determine its distance from p when t=3.
2.
The gradient of a curve at any point (x,y) is
3
x
2
. Given that the curve passes through the point (-2,3), find its equation.
3.
Evaluate
∫
3
-
1
(
2
x
+
3
)
d
x
4.
The velocity v m/s of a particle moving along a straight line at any time t(sec) is given by v=3t-2. Its distance x(m) at the time t=0 is equal to 2. Calculate x when t=4
5.
The velocity of a particle moving in a straight line after t seconds is given by
v
=
6
t
-
t
2
+
4
m/s. Calculate (a) The acceleration of the particle after 2 seconds (b) The distance covered by the particle between t=2 second and t= 5 seconds (c) The time when the particle will be momentarily at rest.
6.
A particle moves on a straight line. The velocity after t second is given by
v
=
3
t
2
-
6
t
-
8
. The distance of the particle from the origin after one second is 10 metres. Calculate the distance of the from the origin after 2 seconds.
7.
The acceleration a m/
s
2
of a particle moving In a straight line is given by a=18t-4, where t is time in seconds. The initial velocity of the particle is 2m/s. (a) Find the expression for velocity in terms of t (b) Determine the time when the velocity is again 2 m/s.
8.
(a) The gradient function of a curve is given by
d
x
d
y
=
2
x
2
-
5
. Find the equation of the curve, given that y=3 when x=2. (b) The velocity,v m/s of a moving particle after t seconds is given by
v
=
2
t
3
+
t
2
-
1
. Find the distance covered by the particle in the interval
1
≤
t
≥
3
9.
The diagram below shows a straight line intersecting the curve at the points P and Q. The line also cuts x- axis at (7,0) and y-axis (0,7) (a) Find the equation of the straight line in the form y=mx+c (b) Find the coordinate of P and Q. (c) Calculate the area of the shaded region
10.
The velocity Vm
s
-
1
of a particle in motion is given by
V
=
3
t
2
-
t
+
4
where t is the time in seconds. Calculate the distance travelled by the particle between the time t=1 seconds and t= 5 seconds.
11.
The gradient function of a curve is given by
d
y
d
x
=
x
2
-
8
x
+
2
. If the curve passes through the point ( 0,2) find its equation.
12.
A particle moves in a straight line. It passes through point O at t=0 with velocity v=5m/s. the acceleration a m/
s
2
of the particle at time t seconds after passing through O is given by a=6t+4. (a) Express the velocity v of the particle at time t seconds in terms of t. (b) Calculate: (i) the velocity of the particle when t=3 (ii)the distance covered by the particle between t=2 and t=4
13.
The acceleration a ms-1 of a particle is given by
a
=
25
-
9
t
2
, where t is the time in seconds after the particle passes a fixed point O. if the particle passes O, with a velocity of 4 m
s
-
1
, find (a) An expression for velocity V, in terms of t (b) The velocity of the particle when t= 2 seconds
14.
The gradient of a curve at point (x,y) is 4x-3. The curve has a minimum value of
-
1
8
. (a) Find ;(i) the value of x at the minimum point . (ii) The equation of the curve. (b) P is a point on the curve in part (a) (ii) above. If the gradient of the curve at p is -7, find the coordinates of p.
15.
A particle moving in a straight line passes through a fixed point O with a velocity to 9m/s. the acceleration of the particle, t seconds after passing through O is given by a=(10-2t) m/
s
2
. Find the velocity of the particle when t=3 seconds.
16.
A particle moves in a straight line through point p. its velocity v m/s is given by v=2-t where t is time in seconds, after passing p. The distance s of the particle from p when t=2 is 5 metres. Find the expression for s in terms of t.
17.
A particle moves in a straight line from a fixed point. Its velocity Vm
s
-
1
after t seconds is given by
V
=
9
t
2
-
4
t
+
1
. Calculate the distance travelled by the particle during third second.
18.
A particle moves in a straight line from a fixed point. Its velocity Vm
s
-
1
after t seconds is given by
v
=
3
t
2
-
6
t
-
9
. The figure below is a sketch of the velocity- time graph of the particle. Calculate the distance the particle moves between t=1 and t=4
19.
A particle starts from O and moves in straight line so that its velocity Vm
s
-
1
after time t seconds is given by
V
=
3
t
-
t
2
. The distance of the particle from O at time t seconds is s metres. (a) Express s in terms of t and c where c is a constant. (b) Calculate the time taken before the particle returns to O.
20.
The gradient of curve is given by
d
y
d
x
=
x
2
-
4
x
+
3
. The curve passes through the point (1,0). Find the equation of the curve.
21.
A particle was moving along a straight line. The acceleration of the particle after t seconds was given by (9-3t)m
s
-
2
. The initial velocity of the particle was 7m
s
-
1
. Find: (a) The velocity (v) of the particle at any given time (t) (b) The maximum velocity of the particle (c) The distance covered by the particle by the time it attained maximum velocity.
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