A Post.

In an attempt to further explain Reimann Sums, I shall first give the formula for each Middle RAM, Left RAM, Right RAM, Trapezoidal RAM then explain each in layman’s terms, all while attempting an example problem pertaining to all of the aforementioned sums.

*One must remember that these sums are all ways of estimating the area under a curve (hence the fact that this is under integration). Particular ones work better at estimating certain problems. However, as a general rule, trapezoidal is the most accurate when, and only when, the delta x is not zero.

LRAM-left hand approximation

When doing this, a trick is to start with the furthest left interval (hence the LEFT Reimann sum) and add delta x until right before you reach your b interval term and substitute those numbers into the original function. Then sum all of those values and multiply by delta x. Simply stated:

x [f(x+x) + … + f(b-x)]

RRAM-right hand approximation

This procedure is the same as the left except, you guessed it, you start with the farthest right interval or b!

*note, this formula may not be the same as in your notes, but since addition is commutative, you can start with whatever you like.

x[f(b) + f(b+x)…f(a+x)

MRAM-middle hand approximation

The trick to finding the middle sum is having the capability of finding all of the midpoints in the given interval.

x[f(mid) +…]

You are going to find the numbers in which you find the midpoints of by adding delta x from a to b. (will explain later)

TRAM-trapezoidal approximation

For this particular one, remember that delta x is over two and that you should multiply all of the numbers besides the functions of your intervals by 2.

x[f(a) + 2f(a+x) + 2f(a+2x) +…f(b)]


These are my examples for the above explanation of the four sums.

Given f(x) = x^2 - 3 with 3 subintervals on the interval [1,4], find the LRAM, MRAM, RRAM, and TRAM.

Beginning, one must first find delta x. Delta x is found by subtracting b minus a and dividing by n (subintervals).

So in this case:

x = 4-1/3 = 1

LRAM:

1[f(1) + f(2) + f(3)]

1[-2 + 1 + 6]

5

RRAM:

1[f(2) + f(3) + f(4)]

1[1 + 6 + 13]

20

TRAM:

½ [ f(1) + 2f(2) + 2f(3) + f(4)]

½[ -2 + 2+ 12 + 13]

½ [-1] = -½

MRAM:

1 2 3 4

3/2 5/2 7/2

1[f(3/2 + f(5/2) + f(7/2)]

So after fully breaking down practically step for step the process of finding MRAM, LRAM, RRAM, and TRAM, do you believe you have it?