M131 A,B Calculus I Spring ‘95 Test #7 3/9/95 BJB Name



Math 131 Lab 11: Riemann Sums and the Definite Integral 4/11/06

Purpose: To study the definite integral as the limit of Riemann Sums.

Definition: The definite integral of f from a to b is the limit of the LHS or RHS with n sub-divisions as n gets arbitrarily large, in other words,

[pic][pic] and [pic][pic]. The LHS and RHS are each called Riemann Sums, f is called the integrand, and a and b are called the limits of integration.

Method: Perform the following steps for the definite integrals in 1-3.

a) Set up the LHS and RHS for the case n = 5 by hand. Give each term. (You can use the table on your calculator to get the functional values.)

b) Consider the graph of the integrand on the given interval. Note the points of subdivision for n = 5. Decide from the given graph below if a LHS or RHS is being calculated. Label height and width of each rectangle. Compare this with your result in part a.

c) Use the nintegrl program to visualize how increasing the number of rectangles on a fixed interval converges to the integral. We’ll illustrate the LHS & RHS or 5, 10, 50 and 250 subdivisions with each member of the group using a different n.

d) Setting up a Riemann Sum with n = 10.

Label the points of subdivision for n = 10. Note that [pic].

Write the subdivision, xi, i = 1, 2, . . . , 10, in terms of a, [pic], and i.

Now write RHS = [pic] for your function and n.

The calculator will compute this with the command [pic].

How would you modify the sum to get a LHS?

e) Fill in the chart for the LHS and RHS for 5, 10, 50 and 250 subdivisions letting your calculator perform the computations. Write the appropriate calculator commands for the case n = 250.

f) From part e what is your estimate of the definite integral?

g) Use [2nd] [7] ( and compare with your estimate.

h) In which cases is the “true” value between the LHS and RHS? Why?

i) For which cases does the definite integral give the area bounded by the x-axis and the curve between the limits of integration? Show how you would obtain this area when it is not given immediately by the definite integral. What is the area in each case?

1.) [pic]

|n |5 | 10 | 50 | 250 |

|LHS | | | | |

|RHS | | | | |

n=250

LHS = [pic]

RHS = [pic]

[pic]

2.) [pic]

|n |5 | 10 | 50 | 250 |

|LHS | | | | |

|RHS | | | | |

n=250

LHS = [pic]

RHS = [pic]

[pic]

3.) [pic]

|n |5 | 10 | 50 | 250 |

|LHS | | | | |

|RHS | | | | |

n=250

LHS = [pic]

RHS = [pic]

[pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download