Review - MAT1322-Review 7.7 Improper Integrals Type I(Innite Interval t f(x)dx = lim t a f(x)dx = b a f(x)dx = lim c f(x)dx b f(x)dx c t f(x)dx t f(x)dx

# Review - MAT1322-Review 7.7 Improper Integrals Type...

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MAT1322-Review 7.7 Improper Integrals Type I (Infinite Interval) Z a f ( x ) dx = lim t →∞ Z t a f ( x ) dx, Z b -∞ f ( x ) dx = lim t →∞ Z b t f ( x ) dx, Z -∞ f ( x ) dx = Z c f ( x ) dx + Z c -∞ f ( x ) dx. Type 2 (Discontinuous Integrand) If f ( x ) is continuous on [ a, b ), then Z b a f ( x ) dx = lim t b Z t a f ( x ) dx ; If f ( x ) is continuous on ( a, b ], then Z b a f ( x ) dx = lim t a Z b a f ( x ) dx ; If f ( x ) is discontinuous at c : a < c < b , then Z b a f ( x ) dx = Z c a f ( x ) dx + Z b c f ( x ) dx. Convergent Integral is a finite number. 7.8 Comparison If f ( x ) and g ( x ) are continuous and f ( x ) g ( x ) on x a . Then (i) R a f ( x ) dx is convergent = R a g ( x ) dx is convergent; (ii) R a g ( x ) dx is divergent = R a f ( x ) dx is divergent. 8.1 Area and Volume Area (i) The area A of the region bounded by y = f ( x ), y = g ( x ), x = a , x = b with f ( x ) g ( x ) is 1
(ii) To find the areaAof the region enclosed byy=f(x) andy=g(x): Find theintersections, then use the previous strategy. Volume of solid For a solid S , let A () be the cross-sectional area, then the volume is: V = Z b a A ( x ) dx = Z d c A ( y ) dy. 8.2 Applications to Geometry Volumes of revolution Case 1: If we rotate a region bounded by y = f ( x ), x = a , x = b around the line y = k ( k = 0 ⇐⇒ x-axis) , then A ( x ) = π [ f ( x ) - k ] 2 , V = Z b a π [ f ( x ) - k ] 2 dx. Case 2: If we rotate a region bounded by x = f ( y ), y = c , y = d around the line x = k ( k = 0 ⇐⇒ y-axis) , then A ( y ) = π [ f ( y ) - k ] 2 , V = Z b a π [ f ( y ) - k ] 2 dy. If the function is y = f ( x ), then x = f - 1 ( y ). Case 3. Cylindrical shell: If we rotate a region bounded by y = f ( x ), x = a , x = b about the y -axis, then V = 2 π Z b a xf ( x ) dx. Arc Length Arc length L = Z b a s dx dt 2 + dy dt 2 dt. 2
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9.1 Sequences Sequence: a 1 , a 2 , ..., a n , ... a n is the nth term. If lim n →∞ a n exists, then we say the sequence converges. Otherwise, we say the sequence diverges. Monotonic Sequence Theorem: Every bounded monotonic sequence is convergent. Bound and Convergence : A convergent sequence is bounded, a bounded and mono- tone sequence converges. 9.2 Geometric Series A basic fact is lim n →∞ r n = 0 ⇔ | r | < 1 . Geometric series: if | r | < 1 then j =1 ar j - 1 = a 1 - r . 9.3 Convergence of series Partial sum: S n = n X j =1 a j . Then X j =1 a j = S lim n →∞ S n = S. k n ( n + k ) = 1 n - 1 n + k . Harmonic series j =1 1 n is divergent. If lim n →∞ a n 6 = 0, then j =1 a j is divergent. j =1 ( ca j + db j ) = c j =1 a j + d j =1 b j . 4
Integral test: If f ( x ) is continuous, positive, decreasing, f ( j ) = a j , then X j =1 a j is convergent Z 1 f ( x ) dx is convergent . The p - series: j =1 1 j p is: convergent if p > 1 and divergent if p 1. 9.4 Test for convergence Comparison test: If 0 a j b j , then X j =1 b j is convergent X j =1 a j is convergent . X j =1 a j