4_4_FTC - 4.4 Fundamental Theorem of Calculus Class Notes:...

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Unformatted text preview: 4.4 Fundamental Theorem of Calculus Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: ex: Indefinite Integrals (1 of 22) 4.4 Fundamental Theorem of Calculus Start with Indefinite Integration: complete the following Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: ex: Indefinite Integrals (2 of 22) 4.4 Fundamental Theorem of Calculus Start with Indefinite Integration: complete the following Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: ex: Indefinite Integrals (3 of 22) 4.4 Fundamental Theorem of Calculus Start with the differential of y, dy Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Nov 286:27 PM (4 of 22) 4.4 Fundamental Theorem of Calculus Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Nov 286:27 PM (5 of 22) Estimating Area under a Curve, since y is nonnegative. Also estimating the definite integral. Use multiple trapazoids for better estimate. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Estimating Area, Definite Integral (6 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: FTC (7 of 22) Do not need c, the constant of integration. It gets added and subtracted to add to 0. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: FTC (8 of 22) Compare to approximation using area of trapazoid. Title: Nov 266:22 PM (9 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Homework, Ex. (10 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Homework, Ex. (11 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Example 2 (12 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Example 2 (13 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Examples 3, 4 (14 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Examples 3, 4 (15 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Examples 3, 4 (16 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Examples 3, 4 (17 of 22) Presents some problems. 1. Start with definiton of absolute value and 2. consider what this means regarding the interval from lower to upper limits. Step #1: Absolute value: Step #2: About the interval Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: FTC with Absolute Value (18 of 22) The integrand is defined differently on the interval. Since definite integrals are defined as limits of sums, we replace the original integral with the sum of 2 integrals which have integrands and limits that correspond to the 2 part definition of the original: c a b Rewrite the original integral as the sum of 2 integrals. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: FTC: sums of intervals (19 of 22) a c b Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: FTC with Absolute Value (20 of 22) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: FTC with Absolute Value (21 of 22) Area Slope MVT for Derivatives Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: MVT, Integrals: Area (22 of 22) ...
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This note was uploaded on 09/09/2011 for the course MATH 1431 taught by Professor Any during the Fall '08 term at University of Houston.

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