Question

use the methods of left reimann sum, right reimann sum, trapezoidal rule, and parabolic rule with 8 intervals to approximate:

Answer 1

\[\int\limits_{1}^{3} 1\div(x^2) dx \]

Answer 2

here. got it.

Answer 3

Ok. what's a left Riemann sum?

Answer 4

bugger it, I'll tell. We are partitioning [1,3] into 8 equally sized subintervals, [1,1.25], [1.25, 1.50], ...., [2.75,3].

The left Riemann sum is equal to the value of the function f(x) = 1/x^2 evaluated at the left of each interval times the length of the interval

Answer 5

i.e.,
\[ L_8 = f(1)(1.25-1) + f(1.25)(1.50-1.25) + .... + f(2.75)(3-2.75) \]

Answer 6

clear?

Answer 7

I have no idea what you just did lol... i know a left reimann sum is the area of the rectangles inside a function

Answer 8

i do have all the formulas for them but is there a simper way

Answer 9

Draw a diagram of this function. Mark out carefully and exactly the eight partitions on he x axis. Draw the rectangles.

I have written down the expression for the "left" Riemann sum.

Answer 10

yea i got that

Answer 11

there isn't an 'easier' way. but really once you get this you'll find it's not really so scary after all. just be patient, make sure you know what the definitions really are.

Answer 12

it takes soooo much time

Answer 13

each one of the terms is the height of the triangle times the width.

Not really, because all the widths are the same, so
\[ L_8 = (1/4) (f(1) + f(1.25) + f(1.50) + .... + f(2.75))

Answer 14

\[ L_8 = (1/4) (f(1) + f(1.25) + f(1.50) + .... + f(2.75)) \]

Answer 15

just calculate it. with a calculate or a spreadsheet it should take all of one minute.

Answer 16

calculator.

Answer 17

see my real problem is my calculator won't accept the program for these when everyone else in class can... :( so i have to do these by hand..

Answer 18

hm. Even with a basic calculator and the M+ key this should be doable. But whatever. You might also want to use Excel. that's probably what I'd do for this problem.

Answer 19

didnt' even think of excel!!!!!!!