1

Example 2 Estimate by the six Rectangle Rules using the regular

partition P of the interval [0, ] into 6 subintervals.

Solution Observe that the function with g(0)=1 is continuous at

x=0 because Note that P = {0, /6, /3, /2, 2/3, 5/6, } with

each subinterval of width /6. The six subintervals are:

[0, /6 ], [/6, /3], [/3, /2], [/2, 2/3], [2/3, 5/6] and [5/6, ].

Then

The Lk are the heights of the rectangles used to approximate this definite integral.

dxxx

0

sin

xx

xgsin

)(

. sin

6543210LLLLLL

6dx

xx

In the Left Endpoint Rule the Lk are the values of g on the left endpoints of the six subintervals: g(0), g(/6), g(/3), g(/2), g(2/3), g(5/6).

y

x0

1

.sin

lim 1xx

0x

xx

ysin

2

In the Right Endpoint Rule the Lk are the values of g on the right endpoints of the six subintervals: g(/6), g(/3), g(/2), g(2/3), g(5/6), g().

In the Midpoint Rule the Lk are the values of g on the midpoints of the six subintervals: g(/12), g(/4), g(5/12), g(7/12), g(3/4), g(11/12).

In the Trapezoid Rule the Lk are the averages of the values of g on the endpoints of each of the six subintervals:

½[g(0)+ g(/6)], ½[g(/6)+ g(/3)], ½[g(/3)+ g(/2)],

½[g(/2)+ g(2/3)], ½[g(2/3)+ g(5/6)], ½[g(5/6)+ g()].

Therefore, the Lower Riemann sum coincides with the estimate of the Right Endpoint Rule and the Upper Riemann sum coincides with estimate of the Left Endpoint Rule.

Since the function g is decreasing on [0,1], it has its maximum value at the left endpoint of each subinterval and its minimum value at the right endpoint of each subinterval.

The values of the Lk are summarized in the table on the next slide

The six subintervals are: [0,/6 ], [/6,/3], [/3,/2], [/2, 2/3], [2/3,5/6], [5/6,].

y

x0

1

xx

ysin

3

The Left Endpoint Rule and Upper Riemann Sum give the same estimate:

The Right Endpoint Rule and Lower Riemann Sum give the same estimate:

The Midpoint Rule gives the estimate:

The Trapezoid Rule gives the estimate:

107219141463782795516

LLLLLL6

dxxx

6543210......

sin

LeftEndpoint

Rule

RightEndpoint

Rule

Midpoint Rule

Trapezoid Rule

LowerRiemann

Sum

UpperRiemann

Sum

L1

L2

L3

L4

L5

L6

g(0)=1

g(/6).955

g(/3).827

g(/2).637

g(2/3).414

g(5/6).191

g(/6).955

g(/3).827

g(/2).637

g(2/3).414

g(5/6).191

g()=0

g(/12) .989

g(/4) .900

g(5/12) .738

g(7/12).527

g(3/4) .300

g(11/12) .090

½(1+.955)

½(.955+.827)

½(.827+.637)

½(.637+.414)

½(.414+.191)

½(.191+0)

g(/6).955

g(/3).827

g(/2).637

g(2/3).414

g(5/6).191

g()=0

g(0)=1

g(/6).955

g(/3).827

g(/2).637

g(2/3).414

g(5/6).191

583101914146378279556

LLLLLL6

dxxx

6543210......

sin

85610903005277389009896

LLLLLL6

dxxx

6543210.......

sin

84610963035267328919786

LLLLLL6

dxxx

6543210.......

sin