Final Exam Term 151 Term 142 Improper Integral

Final Exam Term 151 Term 142 Improper Integral and Ch 11 15 16 Others 13 12 Remark: (24) 1) Chapter 11 Term 151 Term 142 Others (Techniques of Integrations) 8 6 Others-Others 4 4 Fund Thm calculs 1 1 2) Improper Integral 3) Techniques of Integration Volume+surface+arc+ area

Final Exam Term 121 Term 112 Improper Integral and Ch 11 16 16 Others 12 12 Remark: (24) 1) Chapter 10 Term 121 Term 112 Others (Techniques of Integrations) 8 8 Others-Others 4 4 2) Improper Integral 3) Techniques of Integration

Final Exam Term 132 Term 112 Improper Integral and Ch 11 15 15 Others 13 13 Remark: (24) 1) Chapter 10 Term 132 Term 112 Others (Techniques of Integrations) 9 11 Others-Others 4 2 2) Improper Integral 3) Techniques of Integration

Final Exam Term 131 Term 112 Improper Integral and Ch 11 14 16 Others 14 12 Remark: (24) 1) Chapter 10 Term 131 Term 112 Others (Techniques of Integrations) 7 8 Others-Others 7 4 2) Improper Integral 3) Techniques of Integration

SEQUENCES How to find a limit of a sequence (convg or divg) (IF you can) Use other prop. use Math-101 to find the limit. To find the limit Example: 1)Sandwich Thm: abs, r^n, bdd+montone Example: 1)Absolute value: 2)Cont. Func. Thm: 2)Power of r: 3)L’Hôpital’s Rule: 3)bdd+montone: Bdd + monton convg

SEQUENCES Example Find Faster

SEQUENCES

SEQUENCES Example Find where Sol: by sandw. limit is 0

SEQUENCES

SEQUENCES

SEQUENCES

Sec 11. 3: THE INTEGRAL TEST AND ESTIMATES OF SUMS Facts about: (Harmonic Seris) 1)The harmonic series diverges, but very slowly. the sum of the first million terms is less than the sum of the first billion terms is less than 15 22 2) If we delete from the harmonic series all terms having the digit 9 in the denominator. The resulting series is convergent.

Sec 11. 3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

THE COMPARISON TESTS THEOREM: (THE LIMIT COMPARISON TEST) both series converge or both diverge. With positive terms and Converge, then and divg, then convg divg

THE INTEGRAL TEST AND ESTIMATES OF SUMS TERM-102

Alternating Series, Absolute and Conditional Convergence THM: Proof: convg

Given that Study: convg

ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-082

VOLUMES BY CYLINDRICAL SHELLS rotating line Parallel to y-axis The volume is given by CYLINDRICAL SHELLS Find the surface area rotating line Parallel to x-axis The volume is given by Find the surface area

VOLUMES BY CYLINDRICAL SHELLS T-111