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![p-Integrals In Theory
1
Consider a p-Integral of the form ⌠ p where p is a positive integer.
dx
1
⌡x
0
1
Case 1: How does ⌠ p behave (converge or diverge) when 0<p<1?
dx
1.1
⌡x
0
1 1 -p+1 1
⌠dx = lim ⌠x-pdx = lim x = lim 1 –a , (1-p)>0
1-p
+⌡ +-p+1 +1-p 1-p
⌡x a 0
p a 0 a a 0
0 a
1
⌠dx = 1 , ∀p∈(0, 1)
⌡ xp 1-p
0
1
⌠dx , 0<p<1, Converges!
∴ p
⌡x
0
1
1.2 Case 2: How does ⌠dx behave (converge or diverge) when p=1?
⌡ xp
0
1 1
1
⌠dx = lim ⌠dx = lim ln(x)] = lim [(ln(1)–ln(a)]
⌡ x p a 0+ ⌡ x a 0 + a a 0
+
0 a
1
∴⌠ p , p=1, Diverges!
dx
⌡x
0
1
Case 3: How does ⌠ p behave (converge or diverge) when p>1?
dx
1.3
⌡x
0
1 1 -p+1 1
⌠dx = lim ⌠x-pdx = lim x = lim 1 –a , (1-p)<0
1-p
+⌡ +-p+1 +1-p 1-p
⌡x a 0
p a 0 a a 0
0 a
1
⌠dx , p>1, Diverges!
∴ p
⌡x
0
a:pintthry.doc 1](https://image.slidesharecdn.com/pintthry-120316090617-phpapp01/85/AP-Calculus-BC-p-int-theory-notes-1-320.jpg)
![p-Integrals In Theory
1
Consider a p-Integral of the form ⌠ p where p is a positive integer.
dx
1
⌡x
0
1
Case 1: How does ⌠ p behave (converge or diverge) when 0<p<1?
dx
1.1
⌡x
0
1 1 -p+1 1
⌠dx = lim ⌠x-pdx = lim x = lim 1 –a , (1-p)>0
1-p
+⌡ +-p+1 +1-p 1-p
⌡x a 0
p a 0 a a 0
0 a
1
⌠dx = 1 , ∀p∈(0, 1)
⌡ xp 1-p
0
1
⌠dx , 0<p<1, Converges!
∴ p
⌡x
0
1
1.2 Case 2: How does ⌠dx behave (converge or diverge) when p=1?
⌡ xp
0
1 1
1
⌠dx = lim ⌠dx = lim ln(x)] = lim [(ln(1)–ln(a)]
⌡ x p a 0+ ⌡ x a 0 + a a 0
+
0 a
1
∴⌠ p , p=1, Diverges!
dx
⌡x
0
1
Case 3: How does ⌠ p behave (converge or diverge) when p>1?
dx
1.3
⌡x
0
1 1 -p+1 1
⌠dx = lim ⌠x-pdx = lim x = lim 1 –a , (1-p)<0
1-p
+⌡ +-p+1 +1-p 1-p
⌡x a 0
p a 0 a a 0
0 a
1
⌠dx , p>1, Diverges!
∴ p
⌡x
0
a:pintthry.doc 1](https://image.slidesharecdn.com/pintthry-120316090617-phpapp01/75/AP-Calculus-BC-p-int-theory-notes-1-2048.jpg)
![p-Integrals In Theory
+∞
∞
Consider a p-Integral of the form ⌠ p where p is a positive integer.
dx
2
⌡x
1
+∞
Case 1: How does ⌠ p behave (converge or diverge) when 0<p<1?
dx
2.1
⌡x
1
+∞ b -p+1 b
⌠dx = lim ⌠x-pdx = lim x = lim b – 1 , (1-p)>0
1-p
⌡ x b +∞
p ⌡ b +∞-p+1 1 b +∞ 1-p 1-p
1 1
+∞
∴ ⌠ p , 0<p<1, Diverges!
dx
⌡x
1
+∞
Case 2: How does ⌠ p behave (converge or diverge) when p=1?
dx
2.2
⌡x
1
+∞ b
b
⌠dx = lim ⌠dx = lim ln(x)] = lim [(ln(b)–ln(1)]
⌡ xp b +∞⌡ x b +∞ 1 b +∞
1 1
+∞
∴ ⌠ p , p=1, Diverges!
dx
⌡x
1
+∞
Case 3: How does ⌠ p behave (converge or diverge) when p>1?
dx
2.3
⌡x
1
+∞ b -p+1 b
⌠dx = lim ⌠x-pdx = lim x = lim b – 1 , (1-p)<0
1-p
⌡ x b +∞
p ⌡ b +∞-p+1 1 b +∞ 1-p 1-p
1 1
+∞
⌠dx = 1 , ∀p∈(1, +∞)
⌡ xp p-1
1
+∞
∴ ⌠ p , p>1, Converges!
dx
⌡x
1
a:pintthry.doc 2](https://image.slidesharecdn.com/pintthry-120316090617-phpapp01/85/AP-Calculus-BC-p-int-theory-notes-2-320.jpg)
![p-Integrals In Theory
3 Conclusions: Convergence Relations
1
3.1 Theorem 3.1: ⌠dx = 1 if and only if 0<p<1.
⌡ xp 1-p
0
1
3.2 Corollary 3.2: ⌠dx Diverges ∀p≥1.
⌡ xp
0
+∞
Theorem 3.3: ⌠ p =
dx 1
3.3 if and only if p>1.
⌡ x p-1
1
+∞
Corollary 3.4: ⌠ p Diverges ∀p∈(0, 1]
dx
3.4
⌡x
1
a:pintthry.doc 3](https://image.slidesharecdn.com/pintthry-120316090617-phpapp01/85/AP-Calculus-BC-p-int-theory-notes-3-320.jpg)
Uploaded byA Jorge Garcia
AP Calculus BC: p-int theory notes
1. p-Integrals of the form ∫x1/p dx are analyzed for convergence based on the value of p. 2. When 0 < p < 1, the integral ∫x1/p dx converges. When p = 1, it diverges, and when p > 1 it also diverges. 3. Similarly for integrals from 1 to ∞, the integral ∫x1/p dx converges if p > 1 and diverges if p is between 0 and 1 inclusive.
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AP Calculus BC: p-int theory notes
- 1. p-Integrals In Theory 1 Consider a p-Integral of the form ⌠ p where p is a positive integer. dx 1 ⌡x 0 1 Case 1: How does ⌠ p behave (converge or diverge) when 0<p<1? dx 1.1 ⌡x 0 1 1 -p+1 1 ⌠dx = lim ⌠x-pdx = lim x = lim 1 –a , (1-p)>0 1-p +⌡ +-p+1 +1-p 1-p ⌡x a 0 p a 0 a a 0 0 a 1 ⌠dx = 1 , ∀p∈(0, 1) ⌡ xp 1-p 0 1 ⌠dx , 0<p<1, Converges! ∴ p ⌡x 0 1 1.2 Case 2: How does ⌠dx behave (converge or diverge) when p=1? ⌡ xp 0 1 1 1 ⌠dx = lim ⌠dx = lim ln(x)] = lim [(ln(1)–ln(a)] ⌡ x p a 0+ ⌡ x a 0 + a a 0 + 0 a 1 ∴⌠ p , p=1, Diverges! dx ⌡x 0 1 Case 3: How does ⌠ p behave (converge or diverge) when p>1? dx 1.3 ⌡x 0 1 1 -p+1 1 ⌠dx = lim ⌠x-pdx = lim x = lim 1 –a , (1-p)<0 1-p +⌡ +-p+1 +1-p 1-p ⌡x a 0 p a 0 a a 0 0 a 1 ⌠dx , p>1, Diverges! ∴ p ⌡x 0 a:pintthry.doc 1
- 2. p-Integrals In Theory +∞ ∞ Consider a p-Integral of the form ⌠ p where p is a positive integer. dx 2 ⌡x 1 +∞ Case 1: How does ⌠ p behave (converge or diverge) when 0<p<1? dx 2.1 ⌡x 1 +∞ b -p+1 b ⌠dx = lim ⌠x-pdx = lim x = lim b – 1 , (1-p)>0 1-p ⌡ x b +∞ p ⌡ b +∞-p+1 1 b +∞ 1-p 1-p 1 1 +∞ ∴ ⌠ p , 0<p<1, Diverges! dx ⌡x 1 +∞ Case 2: How does ⌠ p behave (converge or diverge) when p=1? dx 2.2 ⌡x 1 +∞ b b ⌠dx = lim ⌠dx = lim ln(x)] = lim [(ln(b)–ln(1)] ⌡ xp b +∞⌡ x b +∞ 1 b +∞ 1 1 +∞ ∴ ⌠ p , p=1, Diverges! dx ⌡x 1 +∞ Case 3: How does ⌠ p behave (converge or diverge) when p>1? dx 2.3 ⌡x 1 +∞ b -p+1 b ⌠dx = lim ⌠x-pdx = lim x = lim b – 1 , (1-p)<0 1-p ⌡ x b +∞ p ⌡ b +∞-p+1 1 b +∞ 1-p 1-p 1 1 +∞ ⌠dx = 1 , ∀p∈(1, +∞) ⌡ xp p-1 1 +∞ ∴ ⌠ p , p>1, Converges! dx ⌡x 1 a:pintthry.doc 2
- 3. p-Integrals In Theory 3 Conclusions: Convergence Relations 1 3.1 Theorem 3.1: ⌠dx = 1 if and only if 0<p<1. ⌡ xp 1-p 0 1 3.2 Corollary 3.2: ⌠dx Diverges ∀p≥1. ⌡ xp 0 +∞ Theorem 3.3: ⌠ p = dx 1 3.3 if and only if p>1. ⌡ x p-1 1 +∞ Corollary 3.4: ⌠ p Diverges ∀p∈(0, 1] dx 3.4 ⌡x 1 a:pintthry.doc 3