9.3 Estimate the area under f ( x ) = 1/ x^ 2 +1 on the interval [-1 ; 1] with a

Question

9.3 Estimate the area under f(x) = 1/x^2+1 on the interval [-1; 1] with a right endpoint approxi-

mation using n = 2, 5, and 10 rectangles.

9.4 Suppose the odometer on our car is broken and we want to estimate the distance driven over

a 30-second time interval. We take speedometer readings every ve seconds and record them

as shown in Table 1.

Time (s) 0 , 5, 10, 15, 20, 25, 30

Velocity (mi/h) 17, 21, 24, 20, 32, 31, 28

Table 1: The values show the velocity every ve seconds.

(a) Use a left endpoint approximation to estimate the distance driven.

(b) Use a right endpoint approximation to estimate the distance driven.

Hint: Convert the velocity to ft/s.

Explanation / Answer

9.4) velocity = 1.47fit/sec (1 mile/hour)

  Velocity (fit/sec) 25, 31, 35, 29, 47, 46, 41

x=vt+c ( x=0 , v= 25 fit/sec)

left endpoint appoxi. =

baf(x)dx?Ln=?i=1nf(xi?1)?x.

We have f(x)=integ(vdt) , a=0, b=30, n=6, so ?x=b?an=30-0/6=5

Interval is divided into n=7 subintervals: [0,5], [5,10], [10,15], [15,20], [20,25] , [25,30]

I?Ln=?x(f(0)+f(5)+f(10)+f(15)+f(20)+f(25)+f(30))

= 4[(0)+(155)+(350)+(425)+(940) +(1150)+(1230)]

= 17000fit

Right endpoint approximation gives (right endpoints of intervals are 5, 10, 15, 20, 25 , 30 )

I?Rn=?x(f(5)+f(10)+f(15)+f(20)+f(25)+f(30))=

= 4[+(155)+(350)+(425)+(940) +(1150)+(1230)]

= 17000fit

9.3)

Answer: 2

Hand Steps to derive the Answer -

Height Value:

?x = (1 - -1)/2

?x = 2/2

?x = 1

Generated X values plugged into function:

(1) * ( f(0) + f(1) )

Function values:

(1) * ( + 2 )

Answer: 24.622222222222

Hand Steps to derive the Answer -

Height Value:

?x = (1 - -1)/5

?x = 2/5

?x = 0.4

Generated X values plugged into function:

(0.4) * ( f(-0.6) + f(-0.2) + f(0.2) + f(0.6) + f(1) )

Function values:

(0.4) * ( 3.7777777777778 + 26 + 26 + 3.7777777777778 + 2 )

Answer: 16.236111111111

Hand Steps to derive the Answer -

Height Value:

?x = (1 - -1)/10

?x = 2/10

?x = 0.2

Generated X values plugged into function:

(0.2) * ( f(-0.8) + f(-0.6) + f(-0.4) + f(-0.2) + f(0) + f(0.2) + f(0.4) + f(0.6) + f(0.8) + f(1) )

Function values:

(0.2) * ( 2.5625 + 3.7777777777778 + 7.25 + 26 + + 26 + 7.25 + 3.7777777777778 + 2.5625 + 2 )

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