1

Example 1 Estimate by the six Rectangle Rules using the regular

partition P of the interval [0,1] into 4 subintervals.

Solution This definite integral gives the area of a quarter circle of radius 1 and therefore has value /4. Hence estimating this integral means estimating the value of /4. Note that P = {0, ¼, ½, ¾, 1} with each subinterval of width ¼. The four subintervals are:

[0, ¼ ], [¼, ½], [½,¾] and [¾, 1]

while the function is Then

The Lk are the heights of the 4 rectangles used to approximate this definite integral.

dxx11

02

.)( 2x1xf

. 43211

02 LLLL

41dxx1

In the Left Endpoint Rule the Lk are the values of f on the left endpoints of the four subintervals: f(0), f(¼), f(½), f(¾). In the Right Endpoint Rule the Lk are the values of f on the right endpoints of the four subintervals: f(¼), f(½), f(¾), f(1).

y

x10

2x1y

2

In the Midpoint Rule the Lk are the values of f on the midpoints of the four subintervals: f(1/8), f(3/8), f(5/8), f(7/8).

In the Trapezoid Rule the Lk are the averages of the values of f on the endpoints of each of the four subintervals:

½[f(0)+f(¼)], ½[f(¼)+f(½)], ½[f(½)+f(¾)], ½[f(¾)+f(1)].

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 f 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 4 subintervals are: [0, ¼ ], [¼, ½], [½,¾], [¾, 1].

y

x10

2x1y

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:

874.0661.866.968.141

41 1

4 4321

1

0

2 LLLLdxx

624.00661.866.968.41

41 1

4 4321

1

0

2 LLLLdxx

796048478192799241LLLL

41dxx1

4 43211

02 .....

749.0331.764.917.984.41

41 1

4 4321

1

0

2 LLLLdxx

LeftEndpoint

Rule

RightEndpoint

Rule

Midpoint Rule

Trapezoid Rule

LowerRiemann

Sum

UpperRiemann

SumL1

L2

L3

L4

f(0)=1f(¼).968f(½) .866f(¾) .661

f(¼).968f(½) .866f(¾) .661f(1)=0

f(1/8) .992f(3/8) .927f(5/8) .781f(7/8) .484

½(1+.968)½(.968+.866)½(.866+.661)½(.661+0)

f(¼).968f(½) .866f(¾) .661f(1)=0

f(0)=1f(¼).968f(½) .866f(¾) .661