Chapter 9 Calculus. What is Calculus? Early Results on Areas and Volumes Maxima, Minima, and Tangents The “Arithmetica Infinitorum” of Wallis Newton’s Calculus of Series The Calculus of Leibniz Biographical Notes: Wallis, Newton, and Leibniz. 9.1 What is Calculus?.
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y = xk
Volume of the solid of revolution:
area of cross-section isπr2
and therefore it is required to compute sum
12k + 22k + 32k +… + n2k
n/n = 1
Fermat’s approach(tangent to y = x2)
y = x2
∫01 x1/2dx = 1 - 1/3 = 2/3
∫01 x2dx = 1/3
Expansion of π as infinite product was known to Viète (before Wallis’ discovery):
Series expansion discovered by 15th century Indian mathematicians and rediscovered by Newton, Gregory and Leibniz:
sub. x = 1