Riemann Sums, Integral?

Question

Riemann Sums, Integral?
S[(4/x)+4], top of S is 6, and bottom of Integral is 2

Upper limit of integral:6

Lower limit of Integral:2

Function acted upon ( (4/x) +4)

a) Find the Riemann sum for this integral using the right-hand sums for n=4

(b) Find the Riemann sum for this same integral, using the left-hand sums for n=4

Answer 1

The answer to your problem, as set, is indeterminate, since you DID NOT SPECIFY THE STEP PATTERN TO BE EMPLOYED! (It is simply not sufficient to give n, presumably the NUMBER of subdivision steps.)

It's also interesting to note that the entire concept of (right- or left-)hand Riemann sums should really be called "Newton sums." Newton had already completely covered this, in the first few propositions in his Principia. Riemann simply put into algebraic language the devastatingly simple geometrical concepts and approach that Newton had used, even including the existence of the same limit for both RH and LH sums as the width of the widest arbitrary subdivision tended to zero.

(Admittedly Newton did it for a monotonic function; but without trying hard to define a pathological function for which even Riemann's method wouldn't work, any "run of the mill function" can be stitched together from monotonic sub-parts. Your own function is monotonic, of course.)

No student of integration should be unaware of Newton's beautiful approach; that they may be, is a sad commentary on current day teaching of the subject.

TELL YOUR TEACHER TO CHECK OUT NEWTON"S FIRST FEW PROPOSITIONS! (I think that he/she will be amazed.)

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Postscript for xian gaon, below:

Yes, but if you look at the (standard) basic DEFINITION of a Riemann sum on the Wolfram site, you'll see that the delta x's are ARBITRARY. Although it's not mentioned in the upper box that catches your eye there, there's some (almost) fine print near the foot of the page. That notes that the sums you'll see illustrated by using their tool are for a CONSTANT mesh size only. This underlines my point that to assign a value to ANY Riemann sum, you need to SAY what MESH PATTERN you're assuming, and NOT JUST the number of subdivisions.

Answer 2

a) use x = 3, 4, 5, 6 to get 99/5 = 19.8

b) use x = 2, 3, 4, 5 to get 317/15 = 21.133

The actual integral is 20.3944

Answer 3

Well, i don't know what dr. spock was saying, but if you look on the wolfram site they have a program that will do it for you (and it even makes a cool graph):

http://mathworld.wolfram.com/RiemannSum.html

Answer 4

i hate calculus.
man oh man.