tutorial11.pdf - MATH1013 Calculus I Tutorial 11...

This preview shows page 1 - 6 out of 7 pages.

MATH1013 Calculus ITutorial 11 Anti-derivatives, Reimann Sum and Fundamental Theorem of Calculus1) Antiderivatives 2) Table of Formulas for Indefinite integral
Background image
3) Reimann Sum 4) Special Sums Let?be a positive number? = ? ???=?? =? (?+?)???=???=? (?+?)(??+?)???=???=??(?+?)????=?5) Definite Integral
6) Properties of Integrals (a)∫ ?(?)?? = − ∫ ?(?)??????(b)∫ ?(?)?? = ???(c)∫ ?(?)?? = ∫ ?(?)??????+ ∫?(?)????7) Fundamental Theorem of Calculus Example 1 :Evaluate ∫ √?? + ?????.Example 2:Calculate ????????.8) Integration of Symmetric Functions
Exercises :1)Evaluate the following integrals :(a)∫(???− ???)??(b)∫ (?√??√?) ??(c)∫ (?? ??−??) ??(d)∫ (????−???) ??(e)(?𝒊??? + ?????)??(f) (????𝜽 + ???𝜽 ???𝜽)?𝜽(g)(??? ?𝜽 ??? ?𝜽)?𝜽(h)∫ ??+???(i)∫ √?(???− ?√??)??(j)?+???+????2)Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions.(a)?(?) = ???+ ?? − ?? ; ?(?) = ?(b)?(?) = ? ??? ? ; ?(?) = ?3)Evaluate ?𝐢??→∞{?+? ??+? + ⋯ ??+? } .4)Prove that ?𝐢??→∞? {?𝒊? ??+ ?𝒊? ???+ ⋯ + ?𝒊? (?−?) ??} =?− ??? ??5)Evaluate the following limit by identifying the integral that it represents :?𝐢??→∞∑ [(???)?+ ?]??=?(??)6)Fundamental Theorem of Calculus: (a) ??∫ ?𝒊???????(b) ???∫ √??+ ?????(c)?????? ???????(d)???√? + ???−???(e)????? ?????????(f)???????+?????(g)???(??+ ?)??? ????7)Even and Odd Functions (a)(??? ?) ??𝝅?−𝝅?(b)??−????+????−?8)Integration by Substitutions(a)∫ ??(??+ ??)???(b)?? ???−???, ? > ??+??.

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture