Tunku Abdul Rahman University College, Kuala Lumpur
mathematics 1
ENGINEERIN knf2013

Spring 2016
Tutorial Integration
1. Solve the integral below:
a. 6
b. 3 cos
1
c. sec 2 ()
d.
3 3 +2
3
e.
2. Use integration by parts to find:
a. sin
b. ln
c. ( + 2)e
d.
ln
3
e. cos e
3. Use integration by substitution to find:
a. 2 sin( 2 )
2
b. 3 +1
c.
Tunku Abdul Rahman University College, Kuala Lumpur
mathematics 1
ENGINEERIN knf2013

Spring 2016
Tutorial Integration SOLUTIONS
1. Solve the integral below:
a. 6
b. 3 cos
1
c. sec 2 ()
d.
3 3 +2
3
e.
2. Use integration by parts to find:
a. sin
b. ln
c. ( + 2)e
d.
ln
3
e. cos e
3. Use integration by substitution to find:
a. 2 sin( 2 )
2
b.
Tunku Abdul Rahman University College, Kuala Lumpur
mathematics 1
ENGINEERIN knf2013

Spring 2016
Tutorial Integration Applications
1. A vehicle moves in a straight line along the tracks(axis), starting from the origin
so that after t seconds its velocity (m/s) is given by
= ( 2 5)( 1)
a. Draw the graph of v(t) over the interval t=0 and t=5 seconds
Tunku Abdul Rahman University College, Kuala Lumpur
mathematics 1
ENGINEERIN knf2013

Spring 2016
Tutorial Integration Applications SOLUTIONS
1. A vehicle moves in a straight line along the tracks(axis), starting from the origin
so that after t seconds its velocity (m/s) is given by
= ( 2 5)( 1)
a. Draw the graph of v(t) over the interval t=0 and t=
Tunku Abdul Rahman University College, Kuala Lumpur
mathematics 1
ENGINEERIN knf2013

Spring 2016
Tutorial Integration Applications
1. Sketch the graph whose equation is = 2 12 + 35. Calculate the volume
between the curve and x axis when it is rotated through 2 radians about the axis.
2. A wine cask has a radius at the top of 30 cm and a radius at the
Tunku Abdul Rahman University College, Kuala Lumpur
mathematics 1
ENGINEERIN knf2013

Spring 2016
Tutorial Integration Applications
1. Sketch the graph whose equation is = + . Calculate the volume
between the curve and x axis when it is rotated through radians about the
axis.
Solution:
7
Volume = 5 2
7
= 5 ( 7)2 ( 5)2
7
= 5 cfw_ 4 24 3 + 214 2 840
Tunku Abdul Rahman University College, Kuala Lumpur
mathematics 1
ENGINEERIN knf2013

Spring 2016
Tutorial : VECTOR
1) Given a= (1,1,0) , b =(2,2,1)and c = (0,1,1), evaluate
a) a+b
b) IaI
c) IabI
d) a+1/2(b) +2c
2) A particle P is acted upon by forces(measured in Newton) where F1 = 3i2j+5k, F2 =i+7j3k, F3 =
5ij+4k and F4 =2j+3k. Determine the ma