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Calculus & Summations Problems

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CALCULUS & SUMMATIONS

1.

A)

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2.) Use four rectangles of equal width, attached to the function at their left

endpoint, to estimate the area under f(x) =1/x between x= 1 and x= 5

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CALCULUS & SUMMATIONS 1. A) 3 ∫( √𝑥 + √𝑥 ) 𝑑𝑥 ∫( 𝑥 1/2 + 𝑥 1/3 ) 𝑑𝑥 [ 1 𝑥 2+1 1 +1 2 = + 2𝑥 3/2 3 1 𝑥 3+1 + 𝐶 ] 1 +1 3 + 3𝑥 4/3 4 B) ∫(𝑥 −3 (𝑥 − 1)) 𝑑𝑥 = ∫(𝑥 −2 − 𝑥 −3 ) 𝑑𝑥 +𝐶 𝑥 −2+1 𝑥 −3+1 =[ − ( )] + 𝐶 −2 + 1 −3 + 1 𝑥 −1 = −1 𝑥 −2 − ( )+𝐶 −2 1 1 − +𝐶 2𝑥 2 𝑥 = C) ∫ 4 + √𝑡 𝑑𝑡 𝑡3 = ∫(4𝑡 −3 + 𝑡 −5/2 ) 𝑑𝑡 −5 4𝑡 −3+1 𝑡 2 +1 = + +𝐶 −3 + 1 −5 + 1 2 −3 4𝑡 −2 𝑡2 = + +𝐶 −2 −3/2 = −2𝑡 −2 − 3 −2 2𝑡 +𝐶 3 D) ∫(2𝑒 𝑥 − 3𝑒 −2𝑥 )𝑑𝑥 3𝑒 −2𝑥 = 2𝑒 − ( )+𝐶 −2 𝑥 = 2𝑒 𝑥 + 3 −2𝑥 𝑒 +𝐶 2 2.) Use four rectangles of equal width, attached to the function at their left endpoint, to estimate the area under f(x) =1/x between x= 1 and x= 5 𝑓(𝑥) = 1 2 3 4 1 𝑥 5 As shown in the figure we divide the curve from x=1 to x=5 into 4 rectangles at their left endpoint. 𝑓(𝑥) = 1 𝑥 It can therefore be seen that n=4  x= 5−1 4 =1 L4= f(x0)x + f(x1)x + f(x2)x + f(x3)x L4= f(1)*1 + f(2)*1 + f(3)*1 + f(4)*1 1 1 2 3  L4 = 1 + +  L4 = 2.0833 + 1 4 3. A) 3 𝑘−1 𝑘=1 𝑘 ∑ = 1−1 1 2−1 + 2 1 2 2 3 =0+ + + 3−1 3 = 1.1666 B) 5 ∑ sin 𝑘𝜋 𝑘=1 = sin 𝜋 + sin 2𝜋 + sin 3𝜋 + sin 4𝜋 + sin 5𝜋 =0+0+0+0+0 =0 C) 3 ∑ (? ...
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