Basic Engineering Mathematics, Fifth Edition

334 Basic Engineering Mathematics

Shaded area=

∫π/ 3

0

ydx

=

∫π/ 3

0

sin2xdx=

[

−

1

2

cos2x

]π/ 3

0

=

{

−

1

2

cos

2 π

3

}

−

{

−

1

2

cos0

}

=

{

−

1

2

(

−

1

2

)}

−

{

−

1

2

( 1 )

}

=

1

4

+

1

2

=

3

4

or0.75 square units

Now try the following Practice Exercise

PracticeExercise 141 Area under curves

(answers on page 355)

Unless otherwise stated all answers are in square

units.

1. Show by integration that the area of a rect-
   angle formed by the liney=4, the ordinates
   x=1andx=6andthex-axis is 20 square
   units.


2. Show by integrationthat the area of the trian-
   gle formed by the liney= 2 x, the ordinates
   x=0andx=4andthex-axis is 16 square
   units.
   3. Sketch the curve y= 3 x^2 +1 between
   x=−2andx=4. Determine by integration
   thearea enclosed by thecurve, thex-axis and
   ordinatesx=−1andx=3. Use an approx-
   imate method to find the area and compare
   your result with that obtained by integration.
   4. The forceFnewtons acting on a body at a
   distancexmetres from a fixed point is given

byF= 3 x+ 2 x^2. If work done=

∫x 2

x 1

Fdx,

determine the work done when the body

moves from the position wherex 1 =1mto

that whenx 2 =3m.

Inproblems 5 to9, sketch graphs of thegiven equa-

tions and then find the area enclosed between the

curves, the horizontal axis and the given ordinates.

5. y= 5 x; x= 1 ,x= 4


6. y= 2 x^2 −x+ 1 ; x=− 1 ,x= 2


7. y=2sin2x; x= 0 ,x=

π

4

8. y=5cos3t; t= 0 ,t=

π

6

9. y=(x− 1 )(x− 3 ); x= 0 ,x= 3


10. The velocityvof a vehicletseconds after a
    certain instant is given byv=

(

3 t^2 + 4

)

m/s.

Determine how far it moves in the interval

fromt=1stot=5s.