Steps for finding area using Summation

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Baldric Perry
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Steps for finding area using Summation 1) Identify a o

) out 4) Sum each f(c i ) from i = 1 to n (# of

1 Steps for finding area using Summation 1) Identify a o and a 0 = starting point of the given interval [a, b] where n = # of rectangles 2) Find the c i 's Right: Left: 3) Plug each c i into given f(x) > Simplify f(c i ) out 4) Sum each f(c i ) from i = 1 to n (# of rectangles) Steps for finding area using Summation Feb 7 10:25 AM 1

HOWEVER... width throughout the interval.

Riemann was a German mathematician who is

2 Definition of Area of a Region where we defined and However, c i could exist anywhere within the Δx interval (rectangle) and moreover, Δx does not NEED to be of equal HOWEVER... width throughout the interval. Def of Area of a Region Riemann Sums Georg Riemann was a German mathematician who is credited with generalizing the formula for finding the area under a curve. His generalization allows us to have Δx's within a given interval of varying lengths. Georg Riemann 2

The Norm of the Partition is going to be the notation used to represent the width of the largest subinterval within a given interval [a, b] is called the norm of

b] and let Δ be a partition of [a, b] where Δi is the width of the i th subinterval.

3 The Norm of the Partition is going to be the notation used to represent the width of the largest subinterval within a given interval [a, b] is called the norm of the interval [a, b] Finally, if all the subintervals ARE of equal width, then the partition is regular and Norm Riemann Sum Let f be defined on the interval [a, b] and let Δ be a partition of [a, b] where Δi is the width of the i th subinterval. If ci is any point in the i th subinterval [x i 1, x i ], then the sum is called a Riemann Sum of f for the partition Δ (NOTE: we were doing Riemann Sums last section. For that section, all of the partitions (Δ) were regular Riemann Sum 3

4 As the norm Δ approaches zero (ie: the largest interval gets smaller and smaller and smaller), then the number of rectangles (n) approaches infinity Definite Integral If f is defined on [a, b] and the limit on the Riemann sums over partitions Δ exist, then f is integrable on [a, b] and the limit is denoted by: Riemann Sum Definite Integral This integral, is different from the integrals we did a few days ago. This integral Side is defined Note within a given interval and has a specified value. The integrals we were doing previously were generalized functions. "a" is the lower limit of integration "b" is the upper limit of integration. "a" and "b" correspond to the interval we are evaluating. "c i " is any point within the i'th subinterval Δx i is the width of the i'th subinterval (recall, the Δx's no longer need to be a set width) Interpreting the Definite Integral 4

2 last notes Example: From Definite Integral Def: Definition of Definite Integral Definition of the limit in terms of "n" Recall, as the norm, Δ > 0, the number

5 Two final notes before we do examples: 1) Δx i no longer needs to be a constant width, however, for calculation purposes, we will be defining it as such. 2) c i does not need to be the left endpoint of the interval or the right endpoint, but again, for calculation purposes, we will define it that way. 2 last notes Example: From Definite Integral Def: Definition of Definite Integral Definition of the limit in terms of "n" Recall, as the norm, Δ > 0, the number of rectangles, n > For calculation purposes, we are going to define Δx, so that it is non varying Δx Finally, c i is defined in terms of c the lower bound ( 2) plus Δx i Example #1 5

6 Example: We've defined: So, the area of the graph f(x) = 2x is " 3" in the interval [ 2, 1]. The area is calculated as "negative" because we are below the x axis. Answer #1 Short Cuts! Finding the area of this does not necessarily require us to use a Riemann Sum. Area of Triangle: Short Cut 6

.. integral, then values of those integrals are opposites.

7 Definitions of 2 Special Cases: 1. ie: The area from a point on the graph In Other Words... that is 0 units wide, is ZERO! 2. ie: If you interchange the bounds of the In Other Words... integral, then values of those integrals are opposites. Special Integrals Properties of Definite Integrals: An interval can be broken up into subsections to make it easier. 3. Properties of Indefinite Integrals 7

Here, we are looking at the area under a parabola

Feb 1 6:56 AM Homework: p278 (#3-7, 9-13 odd, 17,

8 Here, we are looking at the area under a parabola - not a familiar figure we know the area to, so we cannot use a short cut, we must use a Riemann Sum and the limit definition. Feb 1 6:56 AM Homework: p278 (#3-7, 9-13 odd, 17, 19, 23, 27, 31, 33, 37, 41, 43) Mid Chapter Test next class Homework 8

RAMs.notebook December 04, 2017

RAMs.notebook December 04, 2017 RAMsnotebook December 04, 2017 Riemann Sums and Definite Integrals Estimate the shaded area Area between a curve and the x-axis How can you improve your estimate? Suppose f(x) 0 x [a, b], then the area