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All my math notes, now in Markdown.

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5-1 Estimating with Finite Sums

Learning Targets

You will be able to

Left and Right Hand Sums

Concepts / Definitions

Definition of a Riemann Sum

Riemann sums are summing areas of rectangles under a curve in a systematic manner.

Rectangles Under the Curve

LRAM Left Rectangle approximation

LRAM Left Rectangle

Calculate the area, for $f(x)$, for the interval $[0,2]$ with 4 four rectangles ($n=4$), using LRAM. The area is approximated with the sum of the four rectangles.

\(LRAM = wh_1 + wh_2 + wh_3 + wh_4\) \(Area \approx \Delta xf(x_1) + \Delta xf(x_2) + \Delta xf(x_3) + \Delta xf(x_4)\) \(\approx \Delta x [f(x_1) + f(x_2) + f(x_3) + f(x_4)]\)

RRAM Right Rectangle approximation

RRAM Right Rectangle

Calculate the area, for $f(x)$, for the interval $[0,2]$ with 4 four rectangles ($n=4$), using RRAM. The area is approximated with the sum of the four rectangles.

Sigma Notation

\[A = \sum_{i=1}^n\Delta x\ f(x_i)\]

Sigma Notation

Write long hand form $\Delta x (f(x_1) + f(x_2) + f(x_3) + … + f(x_n))$

Formula to use on calculator for adding many areas $sum(seq(f(x), x, x_1, x_n, \Delta x) \bullet \Delta x)$

Ways to make estimates better

Many Rectangles - Less Error

Many Rectangles

Midpoint Method

Midpoint Method

Rectangle Approximation Method (RAM)

Since the height of the rectangle varies along the subinterval, in order to find the area of the rectangle, we must use either the left hand endpoint (LRAM) to find the height, the right hand endpoint (RRAM), or the midpoint (MRAM).

The more rectangles you make, the better the approximation.

LRAM

RRAM

MRAM