Winter wk 2 – Thus.13.Jan.05

• Review Calculus Ch.5.1-2: Distance traveled and the Definite integral

• Ch.5.3: Definite integral of a rate = total change

• Ch.5.4: Theorems about definite integrals

Energy Systems, EJZ

Review 5.1: Measuring distance traveled

Speed = distance/time = rate of change of positionv = dx/dt = x/tPlot speed vs time

Estimate x=vt for each intervalArea under v(t) curve = total displacement

Area under curve: Riemann sums

Time interval = total time/number of steps

t = (b-a) / n

Speed at a given time ti = v(ti)

Area of speed*time interval = distance = v(t)*t

Total distance traveled = sum over all intervals

tot 1 2 n

n

ii=0

x v(t ) t + v(t ) t +...+ v(t ) t

= v(t ) t

Calc Ch.5-3 Conceptest

Calc Ch.5-3 Conceptest soln

Areas and Averages

To precisely calculate total distance traveled xtot

take infinitesimally small time intervals: t 0,

in an infinite number of tiny intervals: n n

in

i=0

= lim f(x ) ( )b

a

Areaunder f curve x f x dx

Practice: 5.3 #3, 4, 8 (Ex.5 p.240), 29

Ex: Problem 5.2 #20

Ex: Problem 5.2 #20

Practice: Ch.5.4 #2

Analytic integration is easier

Riemann sums = approximate:

The more exact the calculation, the more tedious.

Analytic = exact, quick, and elegant

Trick: notice that

( )b b

a a

dxv t dt dt

dt

n

tot ini=0

x lim v(t ) t

Analytic integration

Total change in position

x ( ) 'b b b

b a

a a a

dxx x v t dt dt x dt

dt

Trick: Look at your integrand, v.

Find a function of t you can differentiate to get v.

That’s your solution, x!

Ex: if v=t2, then find an x for which dx/dt= t2

Recall: so and x= __2( )

__d t

tdt

1( )p pd tpt

dt

Practice analytic integration

Total change in F = integral of rate of change of F

1. Look at your integrand, f.

2. Find a function of x you can differentiate to get f.

3. That’s your solution, F!

3

______

dFx

dxF

3 dFF f dx x dx dx

dx

Symmetry simplifies some integrals

Practice: Ch.5.4 # 16

Thm: Adding intervals

Calc Ch.5-4 Conceptest

Calc Ch.5-4 Conceptest soln

Thm: Adding & multiplying integrals

Practice: Ch.5.4 # 4, 8

Thm: Max and min of integrals

Practice: Ch.5.4 #