Exam 2 Review - Exam 2 Review(Sections 7.8 8.1 8.2 9.3 9.4 10.1 7.8 Improper Integrals Two types Infinite Intervals 0 0 Ex [email protected] 1 xe x dx lim Z xe x dx s

# Exam 2 Review - Exam 2 Review(Sections 7.8 8.1 8.2 9.3 9.4...

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Exam 2 Review (Sections 7.8, 8.1, 8.2, 9.3, 9.4, 10.1) 7.8 – Improper Integrals Two types Infinite Intervals Ex: Z @1 0 xe x dx = lim s Q @1 Z s 0 xe x dx Ex: Z @1 1 1 1 + x 2 ffffffffffffffffff dx = Z @1 0 1 1 + x 2 ffffffffffffffffff dx + Z 0 1 1 1 + x 2 ffffffffffffffffff dx Then take the limits of both integrals and solve Discontinuity within Bounds Ex: Z 2 5 1 x @ 2 p wwwwwwwwwwwwwwwwwwwwwwww fffffffffffffffffffffffff dx = lim t Q 2 + Z t 5 1 x @ 2 p wwwwwwwwwwwwwwwwwwwwwwww fffffffffffffffffffffffff dx Ex: Z 0 3 dx x @ 1 ffffffffffffffff = Z 0 1 dx x @ 1 ffffffffffffffff + Z 1 3 dx x @ 1 ffffffffffffffff Then take the limits of both integrals and solve Convergence/Divergence If the limits exist, it’s convergent . If any part of the integration yields a non-existent limit, it’s divergent . P-tests Z 1 1 1 x p fffffff dx [ if p > 1 then convergent , if p ≤ 1 then divergent . Z 0 1 1 x p fffffff dx [ if p < 1 then convergent , if p ≥ 1 then divergent . Comparison Test If f(x) is convergent, everything below is convergent. If f(x) is divergent, everything above is divergent.
8.1 – Arc-Length Straight-forward function: L = Z a b 1 + f . x ` a b c 2 s wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww