Chapter 5 | Integration
617
KEY EQUATIONS
• Properties of Sigma Notation ∑ i =1 n c = nc ∑ i =1 n ca i = c ∑ i =1 n a i ∑ i =1 n ⎛ ⎝ a i + b i ⎞ ⎠ = ∑ i =1 n a i + ∑ i =1 n b i ∑ i =1 n ⎛ ⎝ a i − b i ⎞ ⎠ = ∑ i =1 n a i − ∑ i =1 n b i ∑ i =1 n a i = ∑ i =1 m a i + ∑ i = m +1 n a i • Sums and Powers of Integers ∑ i =1 n i =1+2+⋯+ n = n ( n +1) 2 ∑ i =1 n
i 2 =1 2 +2 2 +⋯+ n 2 = n ( n +1)(2 n +1) 6 ∑ i =0 n i 3 =1 3 +2 3 +⋯+ n 3 = n 2 ( n +1) 2 4 • Left-Endpoint Approximation A ≈ L n = f ( x 0 )Δ x + f ( x 1 )Δ x +⋯+ f ( x n −1 )Δ x = ∑ i =1 n
f ( x i −1 )Δ x
• Right-Endpoint Approximation A ≈ R n = f ( x 1 )Δ x + f ( x 2 )Δ x +⋯+ f ( x n )Δ x = ∑ i =1 n
f ( x i )Δ x
• Definite Integral ∫ a b f ( x ) dx = lim n
→∞ ∑ i =1 n
f ⎛
⎞ ⎠ Δ x
⎝ x i *
• Properties of the Definite Integral ∫ a a f ( x ) dx =0 ∫ b a f ( x ) dx =− ∫ a b f ( x ) dx ∫ a b ⎡ ⎣ f ( x )+ g ( x ) ⎤ ⎦ dx = ∫ a b f ( x ) dx + ∫ a b
g ( x ) dx
⌠ ⌡ a b ∫ a ∫ a
⌡ a b
⎦ dx = ⌠
b
f ( x ) dx − ∫
⎡ ⎣ f ( x )− g ( x ) ⎤
g ( x ) dx
a
b cf ( x ) dx = c ∫ a b
f ( x ) for constant c
b
c
f ( x ) dx + ∫ c b
f ( x ) dx = ∫
f ( x ) dx
a
• Mean Value Theorem for Integrals
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