Integration

By Jason Leitmeyer, Shannon Hunt, Doris Bittner

OutlineIndefinite IntegralsDefinite IntegralsUsing Integration to Find Area

◦Total Area◦Net Signed Area

Rectilinear Motion Using Integration◦Position, Velocity, Speed,

Acceleration

Indefinite IntegralsIntegral Power Rule

A constant factor can be moved through an integral sign

cr

xdxx

rr

1

1

dxxfcdxxcf )()(

Indefinite Integrals (cont.)Examples1)

2)

dxxx )1( 3

dxx

xx

4

25 12

Indefinite Integrals (cont.)

dxex

x )32(

dxex x322

3)

4)

5) tdtt 3sin3cos4

Definite Integrals

dxxx9

4

2 *

dxxx32

0

2 1

dxx

5.

5. 21

1

Definite Integrals (cont.)

dxxx 2cos2sin8

0

5

dxx

x 8

0 3

2

9

Rectilinear Motion using integrationFind the position function for a

particle3)0(,sin1)( sttV

v(0)=0 s(0)=0

13ta(t) 2 t

Rectilinear MotionSigned Area – plug into

calculator

24

1

2 x

Total Area-

b

adxxf )( Plug into calculator

Find the total area between the curve Y=sinx and the x-axis over the interval [0,2π]

Computing Displacement and Velocity by Integration

)sin()( tta

10 v

2

2

4

2t

Computing Displacement and Velocity by Integration (cont.)Find the position, velocity, speed

and acceleration at time t=π if v(t)=cos(t) s(0)=1

Computing Displacement and Velocity by Integration (cont.)Find the position function for a

particle

v(0)=0 s(0)=0

13ta(t) 2 t