5 2 Riemann Sums and Area I Riemann

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5. 2 Riemann Sums and Area

5. 2 Riemann Sums and Area

I. Riemann Sums A. ) Let f (x) be defined on [a, b]. Partition

I. Riemann Sums A. ) Let f (x) be defined on [a, b]. Partition [a, b] by choosing These partition [a, b] into n parts of length Δx 1, Δx 2, … Δxn. In each subinterval, choose a point c 1, c 2, …cn and form the sum This is called a Riemann Sum *NOTE: LRAM, MRAM, and RRAM are all Riemann sums.

B. ) Def. - The NORM of P, denoted ||P||, is the length of

B. ) Def. - The NORM of P, denoted ||P||, is the length of the largest subinterval. C. ) Def. – If is defined on [a, b], then the DEFINITE INTEGRAL of f (x) over [a, b] is provided the limit exists. This is denoted *NOTE: If the limit exists, then f (x) is said to be integrable on [a, b].

II. EXISTENCE A. ) Thm: (Existence of the limit of a Riemann Sum) 1.

II. EXISTENCE A. ) Thm: (Existence of the limit of a Riemann Sum) 1. ) If f (x) is continuous on [a, b], then f (x) is integrable on [a, b]. 2. ) If f (x) is bounded and has a finite number of discontinuities on [a, b], then f (x) is integrable on [a, b].

III. Examples A. ) Express the limit as a definite integral: , where P

III. Examples A. ) Express the limit as a definite integral: , where P is a partition of [1, 4].

B. ) Express as a limit. where P is a partition of .

B. ) Express as a limit. where P is a partition of .

III. Definite Integrals and Technology A. ) fn. Int – Under MATH – 9

III. Definite Integrals and Technology A. ) fn. Int – Under MATH – 9 where f (x) is the function, x is the variable of integration, and a and b are the bounds of the integration.

B. ) Use your calculator to evaluate the following definite integrals:

B. ) Use your calculator to evaluate the following definite integrals:

V. Area Under a Curve A. ) Def. - If f (x) ≥ 0

V. Area Under a Curve A. ) Def. - If f (x) ≥ 0 and integrable on [a, b], then the area bounded by the curve, the x-axis, and the vertical lines x = a and x = b is

B. ) Find the area under the curve x = 0 to x =

B. ) Find the area under the curve x = 0 to x = 3. from

C. ) Find the area bounded by the curve the x-axis, and the vertical

C. ) Find the area bounded by the curve the x-axis, and the vertical lines x = -1 to x = 3. By Geometry By Definite Integral - WHY? ? -

D. ) FACT – For any integrable function: **Note – AREA is ALWAYS POSITIVE,

D. ) FACT – For any integrable function: **Note – AREA is ALWAYS POSITIVE, the definite integral MAY be NEGATIVE.

VI. Examples A. ) Determine the value of the following definite integrals by using

VI. Examples A. ) Determine the value of the following definite integrals by using the areas bounded by the graph of the function, the x-axis, and the bounds given.

WHY? ? ?

WHY? ? ?

WHY? ? ?

WHY? ? ?

WHY? ? ?

WHY? ? ?

B. ) Express the area bounded by f (x) and the xaxis in terms

B. ) Express the area bounded by f (x) and the xaxis in terms of an integrable expression and then find it.

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